{ "info": { "id": "ses_f67ae44bbffehttioFiMPV7e59", "slug": "cosmic-cactus", "projectID": "global", "directory": "/private/var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/nudge-opencode-generation-Al18sO", "path": "private/var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/nudge-opencode-generation-Al18sO", "title": "Nudge content generation", "agent": "nudge-generation", "model": { "id": "gpt-6-astra", "providerID": "openai", "variant": "low" }, "version": "1.18.30", "summary": { "additions": 0, "deletions": 0, "files": 0 }, "cost": 0, "tokens": { "input": 23082, "output": 2679, "reasoning": 185, "cache": { "read": 30464, "write": 0 } }, "time": { "created": 1789261888324, "updated": 1789262014598 } }, "messages": [ { "info": { "role": "user", "time": { "created": 1789261888334 }, "agent": "nudge-generation", "model": { "providerID": "openai", "modelID": "gpt-6-astra", "variant": "low" }, "system": "You create Codeforces learning content for Nudge. Your job is to understand the best approach for the problem and teach it clearly in your own words, using existing solutions to inform your reasoning.\n\nStart with a brief, targeted web search for existing solutions so relevant editorials are in context before substantial independent problem-solving. Use any supplied tutorial links first; otherwise search by contest ID and problem index. Prefer the official tutorial/editorial, and read relevant explanations rather than relying on search snippets. Look at accessible alternatives when useful, especially if the official explanation is missing, unclear, or unnecessarily complicated. Accepted submissions can also help clarify an approach. Respect supplied source lookup statuses and avoid repeatedly retrying blocked sources. If no usable source is accessible, proceed from the supplied statement.\n\nTreat existing solutions as a starting point, not a constraint. If an approach is clear and suitable, build on it without unnecessary re-derivation. If an explanation has gaps, seems incorrect, or uses a harder method than necessary, reason through it and derive a better or simpler approach when worthwhile. Choose a correct approach that is easy to teach and fits the constraints. Verify it against the supplied statement and edge cases before writing the hints, editorial, and C++ solution.\n\nThe editorial must be self-contained: explain the key insight, algorithm, why it works, complexity, and implementation details a learner needs, filling in steps the source skips. Keep research and verification concise; spend the output on teaching the solution, not narrating the research process. Follow the user prompt and output schema exactly.", "format": { "type": "json_schema", "schema": { "type": "object", "properties": { "status": { "type": "string", "enum": [ "success", "unsolvable" ], "description": "Set to 'success' when the supplied statement defines a solvable programming problem. Set to 'unsolvable' only if the statement is fundamentally incomplete, contradictory, or depends on an inaccessible required resource." }, "reason": { "anyOf": [ { "type": "string", "minLength": 1, "maxLength": 2000 }, { "type": "null" } ], "description": "If status is 'unsolvable', provide a detailed failure message to show users. Name the concrete statement/resource blocker and any relevant source lookup/access status, such as missing official tutorial, 403/Cloudflare challenge, 404, or unavailable accepted submissions. Otherwise null." }, "hints": { "anyOf": [ { "type": "array", "minItems": 5, "maxItems": 5, "description": "Exactly 5 progressive hints, each building on the last.", "items": { "type": "object", "properties": { "order": { "type": "integer", "minimum": 1, "maximum": 5, "description": "Hint number, 1 through 5." }, "content": { "type": "string", "minLength": 1, "maxLength": 10000, "description": "Markdown hint text." } }, "required": [ "order", "content" ], "additionalProperties": false } }, { "type": "null" } ], "description": "If status is 'success', provide exactly 5 progressive hints. Otherwise null." }, "editorial": { "anyOf": [ { "type": "string", "minLength": 1, "maxLength": 100000 }, { "type": "null" } ], "description": "If status is 'success', a prose editorial explaining the solution in Nudge's own voice, with no links, citations, or source/research notes. Otherwise null." }, "solution": { "anyOf": [ { "type": "string", "minLength": 1, "maxLength": 200000 }, { "type": "null" } ], "description": "If status is 'success', a complete C++ solution. Otherwise null." } }, "required": [ "status", "reason", "hints", "editorial", "solution" ], "additionalProperties": false, "title": "problem_response", "description": "Submit the generated content for the problem, or report if it is unsolvable." }, "retryCount": 2 }, "id": "msg_09851bb4e001vt60m8o1BKX8V3", "sessionID": "ses_f67ae44bbffehttioFiMPV7e59" }, "parts": [ { "type": "text", "text": "Generate Codeforces learning content: five progressive hints, an original editorial, and AC-quality C++.\n\nFollow the editorial-first research workflow in the system instructions. Use the supplied statement to verify that the source solves this exact problem.\n\nWrite clean Markdown with LaTeX as needed. Use quick and clever humor when appropriate. Tell it like it is (don't sugar-coat responses), and use very casual language. You are fully allowed to swear, just don't overdo it like a sailor (be natural). Deconstruct any false assumptions.\n\nGenerate content for Codeforces problem 2259G: \"Index Removal\" (rated 1900).\n\nKnown problem metadata:\n- Contest ID: 2259\n- Index: G\n- Name: Index Removal\n- Rating: 1900 (Codeforces difficulty rating)\n- Tags: binary search, data structures, math\n- Problem URL: https://codeforces.com/contest/2259/problem/G\n\nUse the rating and tags as weak signals only. The supplied statement is the source of truth.\n\nProblem Statement:\n\n
G. Index Removal
time limit per test
2 seconds
memory limit per test
256 megabytes
input
standard input
output
standard output

An array $$$b_1, b_2, \\ldots, b_m$$$ is good if, for each $$$i$$$ $$$(1 \\leq i \\lt m)$$$, $$$b_{i+1} \\geq b_i$$$ and $$$b_{i+1} - b_i \\leq k$$$. An array of length $$$1$$$ is always good.

You are given an initially good array $$$a_1, a_2, \\ldots, a_n$$$. Solve the following problem for each $$$i$$$ $$$(1 \\leq i \\leq n)$$$ independently:

Input

The first line of each input contains $$$t$$$ ($$$1 \\leq t \\leq 10^4$$$) — the number of test cases.

The first line of each test case contains two integers $$$n$$$ and $$$k$$$ ($$$2 \\leq n \\leq 2 \\cdot 10^5, 1 \\leq k \\leq 10^9$$$).

The second line of each test case contains $$$n$$$ integers $$$a_1, a_2, \\ldots, a_n$$$ ($$$1 \\leq a_i \\leq 10^9$$$). It is guaranteed that $$$a$$$ is initially good.

It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$2 \\cdot 10^5$$$.

Output

For each test case, output $$$n$$$ space separated integers on a single line: the $$$i$$$-th integer denoting the solution to the problem for the $$$i$$$-th index.

Example
Input
7
4 2
1 2 4 5
4 1
1 2 3 4
5 7
1 8 9 16 20
5 1000000000
1 6 7 67 6767
6 1
1 1 2 2 3 4
4 1
1 2 3 3
4 2
1 2 4 6
Output
0 1 1 0 
0 2 1 0 
0 2 1 4 0 
0 0 0 0 0 
0 0 0 0 1 0 
0 1 0 0 
0 2 2 0
Note

In the first test case:

In the fourth test case, $$$a$$$ will remain good no matter what element we remove, so the answer for each index is $$$0$$$.

\n\n\n\nSource lookup status:\n\n- Tutorial (en): https://codeforces.com/blog/entry/156457 loaded (200 OK; title: \"Codeforces Round 1119 (Div. 3) Editorial - Codeforces\"; mentions 2259G)\n\n\nGenerate:\n1. Five progressive hints, from a gentle nudge to the key insight.\n2. A self-contained editorial that teaches the solution clearly, including the key insight, algorithm, correctness argument, complexity, and necessary implementation details. Explain non-obvious steps without padding or a discovery-process transcript.\n3. A complete C++26 solution that gets AC on Codeforces.\n\nFormatting & Style Rules:\n- Hints and editorial: valid Markdown with LaTeX as needed (e.g., $dp[i]$, $$\\sum_{i=1}^{n} a_i$$).\n- Hints and editorial must read like Nudge's own explanation. No research notes, source notes, citations, Markdown links, URLs, or references to editorials/submissions/posts.\n- Solution: raw C++ only, no Markdown fences.\n- Keep C++ short, clean, standard, and single-file. Comments are fine when they clarify the idea.\n\nOutput strictness:\n- Return JSON matching the provided schema exactly.\n- If the problem is solvable, return `status: \"success\"`, `reason: null`, and fill in `hints`, `editorial`, and `solution`.\n- Return `status: \"unsolvable\"`, a short `reason`, and set `hints`, `editorial`, and `solution` to null only when the supplied statement is fundamentally incomplete, contradictory, or dependent on an inaccessible resource needed to define the task. Missing research, high rating, uncertainty, or \"can't guarantee AC\" is not a reason to return unsolvable.\n- If you return `status: \"unsolvable\"`, the `reason` is shown directly to users. Be concrete: say what exact statement/resource blocker stopped you, and include relevant source lookup status if it mattered (for example: official tutorial missing, Codeforces returned 403/Cloudflare challenge, tutorial page 404, accepted submissions unavailable).\n- Each hint must be JUST the hint text. No \"Hint 1:\" or subtitles. The UI adds those automatically.\n- Do not start the editorial with an \"# Editorial\" heading. The UI already adds that section. Feel free to add other headers as needed though.\n- Use styling in the editorial if needed! Just that first header is no good but any subheaders or bolding or whatever is good!\n\nFor the C++ solution, you MUST use this template and work around it:\n\n```cpp\n#include \nusing namespace std;\n\nusing ll = long long;\n\nvoid setIO() {\n ios::sync_with_stdio(false);\n cin.tie(nullptr);\n}\n\nint main() {\n setIO();\n}\n```", "id": "prt_09851bb4f001pFcdnN1KRd4v2g", "sessionID": "ses_f67ae44bbffehttioFiMPV7e59", "messageID": "msg_09851bb4e001vt60m8o1BKX8V3" } ] }, { "info": { "parentID": "msg_09851bb4e001vt60m8o1BKX8V3", "role": "assistant", "mode": "nudge-generation", "agent": "nudge-generation", "variant": "low", "path": { "cwd": "/private/var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/nudge-opencode-generation-Al18sO", "root": "/" }, "cost": 0, "tokens": { "total": 9095, "input": 9063, "output": 32, "reasoning": 0, "cache": { "write": 0, "read": 0 } }, "modelID": "gpt-6-astra", "providerID": "openai", "time": { "created": 1789261888446, "completed": 1789261897913 }, "finish": "tool-calls", "id": "msg_09851bbbe001RIUinCJeu2rh2C", "sessionID": "ses_f67ae44bbffehttioFiMPV7e59" }, "parts": [ { "type": "step-start", "id": "prt_09851c662001gVn0sslwC1iDmm", "sessionID": "ses_f67ae44bbffehttioFiMPV7e59", "messageID": "msg_09851bbbe001RIUinCJeu2rh2C" }, { "type": "tool", "tool": "webfetch", "callID": "call_X5lkxcxuorHwVtJ5w8KrH8Ek", "state": { "status": "completed", "input": { "url": "https://codeforces.com/blog/entry/156457", "format": "markdown" }, "output": "---\nimage: https://codeforces.com/codeforces.org/s/95686/images/codeforces-telegram-square-1024x1024.png\ntitle: Codeforces Round 1119 (Div. 3) Editorial - Codeforces\ndescription: Codeforces. Programming competitions and contests, programming community\n---\n\n \n\n \n[![Codeforces](//codeforces.com/codeforces.org/s/95686/images/codeforces-logo-with-telegram-555x130.png \"Codeforces\")](https://codeforces.com/) \n\n[![In English](//codeforces.com/codeforces.org/s/95686/images/flags/24/gb.png \"In English\")](https://codeforces.com/blog/entry/156457?locale=en) [![По-русски](//codeforces.com/codeforces.org/s/95686/images/flags/24/ru.png \"По-русски\")](https://codeforces.com/blog/entry/156457?locale=ru) \n\n[Enter](https://codeforces.com/enter?back=%2Fblog%2Fentry%2F156457)| [Register](https://codeforces.com/register) \n\n \n* [Home](https://codeforces.com/)\n* [Top](https://codeforces.com/top)\n* [Catalog](https://codeforces.com/catalog)\n* [Contests](https://codeforces.com/contests)\n* [Gym](https://codeforces.com/gyms)\n* [Problemset](https://codeforces.com/problemset)\n* [Groups](https://codeforces.com/groups)\n* [Rating](https://codeforces.com/ratings)\n* 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![New comment\\(s\\)](//codeforces.com/codeforces.org/s/95686/images/icons/comment-12x12.png \"New comment(s)\")\n\n| | [Detailed →](https://codeforces.com/recent-actions) |\n| | --------------------------------------------------- |\n\n* [nik\\_exists](https://codeforces.com/profile/nik%5Fexists)\n* [Blog](https://codeforces.com/blog/nik%5Fexists)\n* [Teams](https://codeforces.com/teams/with/nik%5Fexists)\n* [Submissions](https://codeforces.com/submissions/nik%5Fexists)\n* [Groups](https://codeforces.com/groups/with/nik%5Fexists)\n* [Contests](https://codeforces.com/contests/with/nik%5Fexists)\n* [Problemsetting](https://codeforces.com/contests/writer/nik%5Fexists)\n\n### [nik\\_exists's blog](https://codeforces.com/blog/nik%5Fexists)\n\n[ Codeforces Round 1119 (Div. 3) Editorial ](https://codeforces.com/blog/entry/156457) \n\nBy [nik\\_exists](https://codeforces.com/profile/nik%5Fexists \"Expert nik_exists\"), 7 days ago, ![In English](//codeforces.com/codeforces.org/s/95686/images/flags/24/gb.png \"In English\") \n\nThank y'all so much for participating!\n\n[2259A - Moo Language School](https://codeforces.com/contest/2259/problem/A \"Codeforces Round 1119 (Div. 3)\")\n\n**Solution**\n\nTutorial is loading...\n\n**Code**\n\n```\n#include \nusing namespace std;\n\nint main() {\n int t;\n cin >> t;\n\n while (t--) {\n int n, k;\n cin >> n >> k;\n\n string s;\n cin >> s;\n\n int ans = 0;\n for (int i = 0; i < n / k; i++) {\n bool has_non_nhoj_farm = false;\n for (int j = i * k; j < (i + 1) * k; j++) {\n if (s[j] == '0') has_non_nhoj_farm = true;\n }\n ans += !has_non_nhoj_farm;\n }\n\n cout << ans << \"\\n\";\n }\n}\n\n```\n\n**Difficulty**\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Too hard\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") A bit on the harder side\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Good difficulty\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Easy\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Too easy\n\n**Quality**\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Amazing\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Good\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") So-so\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Bad\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Awful\n\n[2259B - Minus Two](https://codeforces.com/contest/2259/problem/B \"Codeforces Round 1119 (Div. 3)\")\n\n**Hint 1**\n\nWhen comparing two numbers that are initially not equal, the only way they can become equal after some number of operations is if at least one of them has an operation performed on it when it is $$$ \\\\lt 2$$$.\n\n**Hint 2**\n\nIf two elements are equal at some point, they'll be equal no matter how many more operations are performed. Therefore, it suffices to look at the end behavior of each element.\n\n**Solution**\n\nTutorial is loading...\n\n**Code**\n\n```\n#include \nusing namespace std;\n \nint main() {\n int t;\n cin >> t;\n \n while (t--) {\n int n;\n cin >> n;\n \n vector vec(n); for (auto &x: vec) cin >> x;\n \n int mod_4[] {0, 0, 0, 0};\n for (auto &x: vec) mod_4[x%4]++;\n cout << max({mod_4[0], mod_4[2], mod_4[1] + mod_4[3]}) << \"\\n\";\n }\n}\n\n```\n\n**Difficulty**\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Too hard\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") A bit on the harder side\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Good difficulty\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Easy\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Too easy\n\n**Quality**\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Amazing\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Good\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") So-so\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Bad\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Awful\n\n[2259C - 101](https://codeforces.com/contest/2259/problem/C \"Codeforces Round 1119 (Div. 3)\")\n\n**Hint 1**\n\nIf a $$$-1$$$ is in between two $$$1$$$s, it would only hurt the answer to make it a $$$1$$$.\n\n**Hint 2**\n\nIf a $$$-1$$$ is not in between two $$$1$$$ s, the $$$-1$$$ would only be included in a subarray satisfying the conditions if it was an endpoint.\n\n**Solution**\n\nTutorial is loading...\n\n**Code**\n\n```\n#include \nusing namespace std;\n \nint main() {\n int t;\n cin >> t;\n \n while (t--) {\n int n;\n cin >> n;\n \n vector vec(n); for (auto &x: vec) cin >> x;\n \n for (int i = 0; i < n; i++) {\n if (vec[i] == -1) vec[i] = 1;\n if (vec[i] == 1) break;\n }\n \n for (int i = n-1; i >= 0; i--) {\n if (vec[i] == -1) vec[i] = 1;\n if (vec[i] == 1) break;\n }\n \n \n for (auto &x: vec) cout << max(x, 0) << \" \";\n cout << \"\\n\";\n }\n}\n\n```\n\n**Difficulty**\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Too hard\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") A bit on the harder side\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Good difficulty\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Easy\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Too easy\n\n**Quality**\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Amazing\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Good\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") So-so\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Bad\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Awful\n\n[2259D - MEX Multiset](https://codeforces.com/contest/2259/problem/D \"Codeforces Round 1119 (Div. 3)\")\n\n**Hint 1**\n\nIf there is only one $$$0$$$ in the array, then exactly one set will have a mex greater than zero\n\n**Hint 2**\n\n$$$MEX(A) = MEX(B), MEX(C) = 0$$$ satisfies the constraints.\n\n**Solution**\n\nTutorial is loading...\n\n**Code**\n\n```\n#include \nusing namespace std;\n \nint main() {\n int t;\n cin >> t;\n \n while (t--) {\n int n;\n cin >> n;\n \n vector vec(n); for (auto &x: vec) cin >> x;\n if (count(vec.begin(), vec.end(), 0) == 1) {\n cout << \"NO\\n\";\n continue;\n }\n cout << \"YES\\n\";\n bool seen_zero = false;\n for (int i = 0; i < n; i++) {\n if (vec[i] != 0) cout << 'A';\n else if (seen_zero) cout << 'B';\n else {\n seen_zero = true;\n cout << 'C';\n }\n }\n cout << \"\\n\";\n }\n}\n\n```\n\n**Difficulty**\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Too hard\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") A bit on the harder side\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Good difficulty\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Easy\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Too easy\n\n**Quality**\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Amazing\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Good\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") So-so\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Bad\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Awful\n\n[2259E - Treasure Map Destruction (Constructive Version)](https://codeforces.com/contest/2259/problem/E \"Codeforces Round 1119 (Div. 3)\")\n\n**Hint 1**\n\nIf an element $$$x$$$ at index $$$i$$$ is not equal to $$$-1$$$, there must be a treasure at either island $$$i + x$$$ or $$$i - x$$$\n\n**Hint 2**\n\nIf an element $$$x$$$ at index $$$i$$$ is not equal to $$$-1$$$, there must no treasures in the exclusive range of indices ($$$i + x$$$, $$$i - x$$$)\n\n**Solution**\n\nTutorial is loading...\n\n**Code**\n\n```\n#include \nusing namespace std;\n#define ll long long\n \nint main() {\n int t;\n cin >> t;\n \n \n while (t--) {\n int n;\n cin >> n;\n \n vector vec(n); for (auto &x: vec) cin >> x;\n \n vector diff(n+1);\n for (int i = 0; i < n; i++) {\n if (vec[i] > 0) {\n diff[max(0, i-vec[i]+1)]++; \n diff[min(n, i+vec[i])]--; \n }\n }\n vector restricted(n);\n int curr_count = 0;\n for (int i = 0; i < n; i++) {\n curr_count += diff[i];\n restricted[i] = curr_count > 0;\n }\n bool possible = true;\n for (int i = 0; i < n && possible; i++) {\n if (vec[i] >= 0) {\n if (i - vec[i] >= 0 && !restricted[i - vec[i]]) continue;\n if (i + vec[i] < n && !restricted[i + vec[i]]) continue;\n possible = false;\n }\n }\n \n if (!possible) cout << -1 << \"\\n\";\n else {\n for (auto x: restricted) cout << !x;\n cout << \"\\n\";\n } \n }\n}\n\n```\n\n**Difficulty**\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Too hard\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") A bit on the harder side\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Good difficulty\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Easy\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Too easy\n\n**Quality**\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Amazing\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Good\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") So-so\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Bad\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Awful\n\n[2259F - Binary Bubble Sort Inversions](https://codeforces.com/contest/2259/problem/F \"Codeforces Round 1119 (Div. 3)\")\n\n**Hint 1**\n\nLook at performing some bubbles and reverse bubbles on small arrays. How many elements are moved?\n\n**Hint 2**\n\nA bubble will result is the leftmost 1 being moved all the way to the right. Similarly, a reverse bubble will result in the rightmost 0 being moved all the way to the left.\n\n**Hint 3**\n\nSince prefix zeroes and suffix ones don't matter, if we continuously remove them, then a bubble removes the last element of the array and a reverse bubble removes the first element of the array. Is there a way to remove the first and last elements of an array in constant time?\n\n**Solution**\n\nTutorial is loading...\n\n**Code**\n\n```\n#include \nusing namespace std;\n \n#define ll long long\nint main() {\n int t;\n cin >> t;\n \n while (t--) {\n int n;\n cin >> n;\n \n vector vec(n); for (auto &x: vec) cin >> x;\n string s;\n cin >> s;\n \n ll inversions = 0;\n int cnt_ones = 0;\n for (int i = 0; i < n; i++) {\n cnt_ones += vec[i];\n if (vec[i] == 0) inversions += cnt_ones;\n }\n cout << inversions << \" \";\n \n deque dq;\n for (auto &x: vec) dq.push_back(x);\n int cnt_zeroes = count(vec.begin(), vec.end(), 0);\n for (int i = 0; i < n; i++) {\n while (dq.size() > 0 && dq.front() == 0) {\n dq.pop_front();\n cnt_zeroes--;\n }\n while (dq.size() > 0 && dq.back() == 1) dq.pop_back();\n if (dq.size() == 0) {\n cout << 0 << \" \";\n continue;\n }\n if (s[i] == '0') {\n inversions -= (dq.size() - cnt_zeroes);\n dq.pop_back();\n cnt_zeroes--;\n }\n else {\n inversions -= cnt_zeroes;\n dq.pop_front();\n }\n cout << inversions << \" \";\n }\n cout << \"\\n\";\n }\n}\n\n```\n\n**Difficulty**\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Too hard\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") A bit on the harder side\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Good difficulty\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Easy\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Too easy\n\n**Quality**\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Amazing\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Good\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") So-so\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Bad\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Awful\n\n[2259G - Index Removal](https://codeforces.com/contest/2259/problem/G \"Codeforces Round 1119 (Div. 3)\") \n\n**Hint 1**\n\nIgnoring indexes 1 and $$$n$$$, if we remove index $$$i$$$, if we perform an operation on index $$$j$$$, all elements between $$$\\[i, j\\]$$$ will also have an operation performed on them.\n\n**Hint 2**\n\nIf we remove index $$$i$$$, and we have to reduce the element at index $$$i+1$$$, $$$i+2$$$, etc, what will we subtract it to?\n\n**Solution**\n\nTutorial is loading...\n\n**Code**\n\n```\n#include \nusing namespace std;\n \n#define ll long long\nint main() {\n int t;\n cin >> t;\n \n while (t--) {\n ll n, k;\n cin >> n >> k;\n \n vector vec(n); for (auto &x: vec) cin >> x;\n vector vec_2(n); for (int i = 0; i < n; i++) vec_2[i] = vec[i] - i * k;\n \n vector prefix(n); prefix[0] = vec[0];\n for (int i = 1; i < n; i++) prefix[i] = prefix[i-1] + vec[i];\n \n vector ans(n);\n for (int i = 1; i < n-1; i++) {\n int lo = i+1;\n int hi = n-1;\n while (lo <= hi) {\n int mid = lo + (hi - lo) / 2;\n if (vec_2[mid] <= (vec[i-1] - i*k)) hi = mid - 1;\n else lo = mid + 1;\n }\n \n ll num_needed = (lo - i - 1);\n ans[i] = (prefix[num_needed + i] - prefix[i]) - (k * (num_needed * (num_needed + 1))/2) - vec[i-1] * num_needed;\n }\n for (auto &x: ans) cout << x << \" \";\n cout << \"\\n\";\n }\n}\n\n```\n\n**Difficulty**\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Too hard\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") A bit on the harder side\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Good difficulty\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Easy\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Too easy\n\n**Quality**\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Amazing\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Good\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") So-so\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Bad\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Awful\n\n[2259H - Treasure Map Destruction (Counting Version)](https://codeforces.com/contest/2259/problem/H \"Codeforces Round 1119 (Div. 3)\")\n\nRead the solution to problem E first\n\n**Hint 1**\n\nLet's consider all locations that we have not restricted: we can separate them into 3 categories: the position is forced to be a treasure, the position has no undestroyed that depend on it, and all other positions. \n\n**Hint 2**\n\nThe islands that are considered \"non-restricted\" form a chain of locations, with each non-restricted chain having a treasure between them. What must be true of the locations in this chain such that no condition is violated?\n\n**Hint 3**\n\nFor every pair of adjacent locations, there must be at least one treasure at one of those locations.\n\n**Solution**\n\nTutorial is loading...\n\n**Code**\n\n```\n#include \nusing namespace std;\n#define ll long long\n \nconst ll MOD = 1000000007;\nconst int MAX_N = 200055;\nint main() {\n int t;\n cin >> t;\n \n \n vector fibo(MAX_N);\n fibo[1] = 1;\n fibo[2] = 1;\n for (int i = 3; i < MAX_N; i++) {\n fibo[i] = (fibo[i-1] + fibo[i-2]) % MOD;\n }\n \n while (t--) {\n int n;\n cin >> n;\n \n vector vec(n); for (auto &x: vec) cin >> x;\n \n vector diff(n+1);\n for (int i = 0; i < n; i++) {\n if (vec[i] > 0) {\n diff[max(0, i-vec[i]+1)]++; \n diff[min(n, i+vec[i])]--; \n }\n }\n \n vector restricted(n);\n vector forced(n);\n vector non_required;\n int curr_count = 0;\n for (int i = 0; i < n; i++) {\n curr_count += diff[i];\n restricted[i] = curr_count > 0;\n }\n \n bool possible = true;\n for (int i = 0; i < n; i++) {\n if (vec[i] != -1) {\n int cnt = 0;\n cnt += (i - vec[i] >= 0 && !restricted[i - vec[i]]); \n cnt += (i + vec[i] < n && !restricted[i + vec[i]]);\n if (cnt == 0) { possible = false;}\n if (cnt == 1 || vec[i] == 0) {\n if (i + vec[i] < n) forced[i + vec[i]] = true;\n if (i - vec[i] >= 0) forced[i - vec[i]] = true;\n }\n }\n }\n for (int i = 0; i < n; i++) {\n if (!restricted[i] && !forced[i]) non_required.push_back(i);\n }\n if (!possible) {\n cout << 0 << \"\\n\";\n continue;\n }\n \n ll total = 1;\n int chain_len = 1;\n for (int i = 0; i < non_required.size(); i++) {\n bool cond1 = i == non_required.size() - 1 || vec[(non_required[i] + non_required[i+1])/2] == (non_required[i+1] - non_required[i])/2;\n bool cond2 = i == non_required.size() - 1 || (non_required[i+1] - non_required[i]) % 2 == 0;\n //if (i != 1) cout << (non_required[i+1] - non_required[i]) << \"\\n\";\n if (i == non_required.size() - 1 || !cond1 || !cond2) {\n //cout << chain_len << \"\\n\";\n total *= fibo[chain_len+2];\n total %= MOD;\n chain_len = 1;\n }\n else chain_len++;\n }\n if (count(vec.begin(), vec.end(), -1) == n) {\n total = (total + MOD - 1) % MOD;\n }\n cout << total << \"\\n\";\n }\n}\n\n```\n\n**Difficulty**\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Too hard\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") A bit on the harder side\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Good difficulty\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Easy\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Too easy\n\n**Quality**\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Amazing\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Good\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") So-so\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Bad\n\n[ ](https://codeforces.com/data/like?action=like \"I like this\") Awful\n\n![](//codeforces.com/codeforces.org/s/95686/images/icons/paperclip-16x16.png) Tutorial of [Codeforces Round 1119 (Div. 3)](https://codeforces.com/contest/2259) \n\n* [![Vote: I like it](//codeforces.com/codeforces.org/s/95686/images/actions/voteup-gray.png \"Vote: I like it\")](#)\n* +186\n* [![Vote: I do not like it](//codeforces.com/codeforces.org/s/95686/images/actions/votedown-gray.png \"Vote: I do not like it\")](#)\n\n* [![Author](//codeforces.com/codeforces.org/s/95686/images/blog/user_16x16.png \"Author\")](https://codeforces.com/profile/nik%5Fexists) [nik\\_exists ](https://codeforces.com/profile/nik%5Fexists)\n* ![Publication date](//codeforces.com/codeforces.org/s/95686/images/blog/date_16x16.png \"Publication date\") 7 days ago\n* [![Comments](//codeforces.com/codeforces.org/s/95686/images/blog/comments_16x16.png \"Comments\")](https://codeforces.com/blog/entry/156457#comments) [116 ](https://codeforces.com/blog/entry/156457#comments)\n\n \n![Comments](//codeforces.com/codeforces.org/s/95686/images/icons/comments-48x48.png \"Comments\") Comments (116) \n\n[Write comment?](#) \n\n| » [nik\\_exists](https://codeforces.com/profile/nik%5Fexists \"Expert nik_exists\") | 7 days ago, [show (+1)](#) # |\n| -------------------------------------------------------------------------------- | ---------------------------- |\n\n| » [ ![](https://codeforces.com/userpic.codeforces.org/no-avatar.jpg) ](https://codeforces.com/profile/nik%5Fexists) [nik\\_exists](https://codeforces.com/profile/nik%5Fexists \"Expert nik_exists\") | 7 days ago, [hide](#) [#](https://codeforces.com/blog/entry/156457?#comment-1390585 \"Link to comment\") \\| [![Vote: I like it](//codeforces.com/codeforces.org/s/95686/images/actions/comment-voteup-blue.png \"Vote: I like it\")](#) +8 [![Vote: I do not like it](//codeforces.com/codeforces.org/s/95686/images/actions/comment-votedown-blue.png \"Vote: I do not like it\")](#) _Auto comment: topic has been updated by [nik\\_exists](https://codeforces.com/profile/nik%5Fexists \"Expert nikexists\") ([previous revision](https://codeforces.com/topic/157099/en2), [new revision](https://codeforces.com/topic/157099/en3), [compare](https://codeforces.com/topic/157099/diff/en2/en3))._ [→](#) [Reply](#) |\n| -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |\n\n* | » » [wbw121124](https://codeforces.com/profile/wbw121124 \"Specialist wbw121124\") | 7 days ago, [show (+1)](#) # |\n| -------------------------------------------------------------------------------- | ---------------------------- |\n\n| » » [ ![](https://codeforces.com/userpic.codeforces.org/3260004/avatar/bfc5f8da6bb8c097.jpg) ](https://codeforces.com/profile/wbw121124) [wbw121124](https://codeforces.com/profile/wbw121124 \"Specialist wbw121124\") | 7 days ago, [hide](#) [#](https://codeforces.com/blog/entry/156457?#comment-1390753 \"Link to comment\") [^](#comment-1390585 \"Parent comment\") \\| [![Vote: I like it](//codeforces.com/codeforces.org/s/95686/images/actions/comment-voteup-blue.png \"Vote: I like it\")](#) +3 [![Vote: I do not like it](//codeforces.com/codeforces.org/s/95686/images/actions/comment-votedown-blue.png \"Vote: I do not like it\")](#) where are codesit only have cpp [→](#) [Reply](#) |\n| --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |\n\n * | » » » [knkttp](https://codeforces.com/profile/knkttp \"Specialist knkttp\") | 7 days ago, [show](#) # |\n| ------------------------------------------------------------------------- | ----------------------- |\n\n| » » » [ ![](https://codeforces.com/userpic.codeforces.org/5193194/avatar/1319107b0a053e8e.jpg) ](https://codeforces.com/profile/knkttp) [knkttp](https://codeforces.com/profile/knkttp \"Specialist knkttp\") | 7 days ago, [hide](#) [#](https://codeforces.com/blog/entry/156457?#comment-1390760 \"Link to comment\") [^](#comment-1390753 \"Parent comment\") \\| [![Vote: I like it](//codeforces.com/codeforces.org/s/95686/images/actions/comment-voteup-blue.png \"Vote: I like it\")](#) 0 [![Vote: I do not like it](//codeforces.com/codeforces.org/s/95686/images/actions/comment-votedown-blue.png \"Vote: I do not like it\")](#) Wait until the hack end [→](#) [Reply](#) |\n| ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |\n * | » » » [nik\\_exists](https://codeforces.com/profile/nik%5Fexists \"Expert nik_exists\") | 6 days ago, [show (+1)](#) # |\n| ------------------------------------------------------------------------------------ | ---------------------------- |\n\n| » » » [ ![](https://codeforces.com/userpic.codeforces.org/no-avatar.jpg) ](https://codeforces.com/profile/nik%5Fexists) [nik\\_exists](https://codeforces.com/profile/nik%5Fexists \"Expert nik_exists\") | 6 days ago, [hide](#) [#](https://codeforces.com/blog/entry/156457?#comment-1390952 \"Link to comment\") [^](#comment-1390753 \"Parent comment\") \\| [![Vote: I like it](//codeforces.com/codeforces.org/s/95686/images/actions/comment-voteup-blue.png \"Vote: I like it\")](#) 0 [![Vote: I do not like it](//codeforces.com/codeforces.org/s/95686/images/actions/comment-votedown-blue.png \"Vote: I do not like it\")](#) Code has been added! [→](#) [Reply](#) |\n| ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |\n\n * | » » » » [wbw121124](https://codeforces.com/profile/wbw121124 \"Specialist wbw121124\") | 6 days ago, [show](#) # |\n| ------------------------------------------------------------------------------------ | ----------------------- |\n\n| » » » » [ ![](https://codeforces.com/userpic.codeforces.org/3260004/avatar/bfc5f8da6bb8c097.jpg) ](https://codeforces.com/profile/wbw121124) [wbw121124](https://codeforces.com/profile/wbw121124 \"Specialist wbw121124\") | 6 days ago, [hide](#) [#](https://codeforces.com/blog/entry/156457?#comment-1391001 \"Link to comment\") [^](#comment-1390952 \"Parent comment\") \\| [![Vote: I like it](//codeforces.com/codeforces.org/s/95686/images/actions/comment-voteup-blue.png \"Vote: I like it\")](#) 0 [![Vote: I do not like it](//codeforces.com/codeforces.org/s/95686/images/actions/comment-votedown-blue.png \"Vote: I do not like it\")](#) OK [→](#) [Reply](#) |\n| ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |\n\n| » [furqansuleymanli](https://codeforces.com/profile/furqansuleymanli \"Newbie furqansuleymanli\") | 7 days ago, [show (+1)](#) # |\n| ----------------------------------------------------------------------------------------------- | ---------------------------- |\n\n| » [ ![](https://codeforces.com/userpic.codeforces.org/5563911/avatar/f4984ba07dcb9e51.jpg) ](https://codeforces.com/profile/furqansuleymanli) [furqansuleymanli](https://codeforces.com/profile/furqansuleymanli \"Newbie furqansuleymanli\") | 7 days ago, [hide](#) [#](https://codeforces.com/blog/entry/156457?#comment-1390593 \"Link to comment\") \\| [![Vote: I like it](//codeforces.com/codeforces.org/s/95686/images/actions/comment-voteup-blue.png \"Vote: I like it\")](#) +2 [![Vote: I do not like it](//codeforces.com/codeforces.org/s/95686/images/actions/comment-votedown-blue.png \"Vote: I do not like it\")](#) good contest! [→](#) [Reply](#) |\n| ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |\n\n* | » » [nik\\_exists](https://codeforces.com/profile/nik%5Fexists \"Expert nik_exists\") | 7 days ago, [show](#) # |\n| ---------------------------------------------------------------------------------- | ----------------------- |\n\n| » » [ ![](https://codeforces.com/userpic.codeforces.org/no-avatar.jpg) ](https://codeforces.com/profile/nik%5Fexists) [nik\\_exists](https://codeforces.com/profile/nik%5Fexists \"Expert nik_exists\") | 7 days ago, [hide](#) [#](https://codeforces.com/blog/entry/156457?#comment-1390642 \"Link to comment\") [^](#comment-1390593 \"Parent comment\") \\| [![Vote: I like it](//codeforces.com/codeforces.org/s/95686/images/actions/comment-voteup-blue.png \"Vote: I like it\")](#) +7 [![Vote: I do not like it](//codeforces.com/codeforces.org/s/95686/images/actions/comment-votedown-blue.png \"Vote: I do not like it\")](#) tysm! [→](#) [Reply](#) |\n| ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |\n* | » » [dhiruuu\\_singh26](https://codeforces.com/profile/dhiruuu%5Fsingh26 \"Newbie dhiruuu_singh26\") | 7 days ago, [show](#) # |\n| ------------------------------------------------------------------------------------------------- | ----------------------- |\n\n| » » [ ![](https://codeforces.com/userpic.codeforces.org/4933574/avatar/bca47b478c630dee.jpg) ](https://codeforces.com/profile/dhiruuu%5Fsingh26) [dhiruuu\\_singh26](https://codeforces.com/profile/dhiruuu%5Fsingh26 \"Newbie dhiruuu_singh26\") | 7 days ago, [hide](#) [#](https://codeforces.com/blog/entry/156457?#comment-1390883 \"Link to comment\") [^](#comment-1390593 \"Parent comment\") \\| [![Vote: I like it](//codeforces.com/codeforces.org/s/95686/images/actions/comment-voteup-blue.png \"Vote: I like it\")](#) 0 [![Vote: I do not like it](//codeforces.com/codeforces.org/s/95686/images/actions/comment-votedown-blue.png \"Vote: I do not like it\")](#) TIGER! TIGER!........ [→](#) [Reply](#) |\n| ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |\n* | » » [While.1](https://codeforces.com/profile/While.1 \"Newbie While.1\") | 6 days ago, [show](#) # |\n| ---------------------------------------------------------------------- | ----------------------- |\n\n| » » [ ![](https://codeforces.com/userpic.codeforces.org/5072827/avatar/578ea0266e326443.jpg) ](https://codeforces.com/profile/While.1) [While.1](https://codeforces.com/profile/While.1 \"Newbie While.1\") | 6 days ago, [hide](#) [#](https://codeforces.com/blog/entry/156457?#comment-1390954 \"Link to comment\") [^](#comment-1390593 \"Parent comment\") \\| [![Vote: I like it](//codeforces.com/codeforces.org/s/95686/images/actions/comment-voteup-blue.png \"Vote: I like it\")](#) 0 [![Vote: I do not like it](//codeforces.com/codeforces.org/s/95686/images/actions/comment-votedown-blue.png \"Vote: I do not like it\")](#) felt that problem F was more trickier than problem G! [→](#) [Reply](#) |\n| --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |\n\n| » [why127pan](https://codeforces.com/profile/why127pan \"Newbie why127pan\") | 7 days ago, [show (+1)](#) # |\n| -------------------------------------------------------------------------- | ---------------------------- |\n\n| » [ ![](https://codeforces.com/userpic.codeforces.org/5327650/avatar/5df2e47dfbff541b.jpg) ](https://codeforces.com/profile/why127pan) [why127pan](https://codeforces.com/profile/why127pan \"Newbie why127pan\") | 7 days ago, [hide](#) [#](https://codeforces.com/blog/entry/156457?#comment-1390594 \"Link to comment\") \\| [![Vote: I like it](//codeforces.com/codeforces.org/s/95686/images/actions/comment-voteup-blue.png \"Vote: I like it\")](#) 0 [![Vote: I do not like it](//codeforces.com/codeforces.org/s/95686/images/actions/comment-votedown-blue.png \"Vote: I do not like it\")](#) nice round thanks ! [→](#) [Reply](#) |\n| --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |\n\n* | » » [nik\\_exists](https://codeforces.com/profile/nik%5Fexists \"Expert nik_exists\") | 7 days ago, [show](#) # |\n| ---------------------------------------------------------------------------------- | ----------------------- |\n\n| » » [ ![](https://codeforces.com/userpic.codeforces.org/no-avatar.jpg) ](https://codeforces.com/profile/nik%5Fexists) [nik\\_exists](https://codeforces.com/profile/nik%5Fexists \"Expert nik_exists\") | 7 days ago, [hide](#) [#](https://codeforces.com/blog/entry/156457?#comment-1390645 \"Link to comment\") [^](#comment-1390594 \"Parent comment\") \\| [![Vote: I like it](//codeforces.com/codeforces.org/s/95686/images/actions/comment-voteup-blue.png \"Vote: I like it\")](#) +6 [![Vote: I do not like it](//codeforces.com/codeforces.org/s/95686/images/actions/comment-votedown-blue.png \"Vote: I do not like it\")](#) tysm! [→](#) [Reply](#) |\n| ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |\n\n| » [megahertz13](https://codeforces.com/profile/megahertz13 \"Pupil megahertz13\") | 7 days ago, [show (+1)](#) # |\n| ------------------------------------------------------------------------------- | ---------------------------- |\n\n| » [ ![](https://codeforces.com/userpic.codeforces.org/2827759/avatar/2d153e90dd0538bd.jpg) ](https://codeforces.com/profile/megahertz13) [megahertz13](https://codeforces.com/profile/megahertz13 \"Pupil megahertz13\") | 7 days ago, [hide](#) [#](https://codeforces.com/blog/entry/156457?#comment-1390596 \"Link to comment\") \\| [![Vote: I like it](//codeforces.com/codeforces.org/s/95686/images/actions/comment-voteup-blue.png \"Vote: I like it\")](#) 0 [![Vote: I do not like it](//codeforces.com/codeforces.org/s/95686/images/actions/comment-votedown-blue.png \"Vote: I do not like it\")](#) good problems, C was kind of annoying to implement [→](#) [Reply](#) |\n\n...196000 bytes truncated...\n\nThe tool call succeeded but the output was truncated. Full output saved to: /Users/zayd/.local/share/opencode/tool-output/tool_09851e0b00010PIdcYFa9lQs1o\nUse the Task tool to have explore agent process this file with Grep and Read (with offset/limit). Do NOT read the full file yourself - delegate to save context.", "metadata": { "truncated": true, "outputPath": "/Users/zayd/.local/share/opencode/tool-output/tool_09851e0b00010PIdcYFa9lQs1o" }, "title": "https://codeforces.com/blog/entry/156457 (text/markdown; charset=utf-8)", "time": { "start": 1789261891966, "end": 1789261897906 } }, "metadata": { "openai": { "itemId": "fc_003f04802b3670cf016aa5f8432d1487d1bc8de1f9c16c3ad1" } }, "id": "prt_09851c665001avnQF9YLi1zHSC", "sessionID": "ses_f67ae44bbffehttioFiMPV7e59", "messageID": "msg_09851bbbe001RIUinCJeu2rh2C" }, { "reason": "tool-calls", "type": "step-finish", "tokens": { "total": 9095, "input": 9063, "output": 32, "reasoning": 0, "cache": { "write": 0, "read": 0 } }, "cost": 0, "id": "prt_09851e0b5001LPxoTjzct9c5vt", "sessionID": "ses_f67ae44bbffehttioFiMPV7e59", "messageID": "msg_09851bbbe001RIUinCJeu2rh2C" } ] }, { "info": { "parentID": "msg_09851bb4e001vt60m8o1BKX8V3", "role": "assistant", "mode": "nudge-generation", "agent": "nudge-generation", "variant": "low", "path": { "cwd": "/private/var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/nudge-opencode-generation-Al18sO", "root": "/" }, "cost": 0, "tokens": { "total": 22494, "input": 13134, "output": 343, "reasoning": 185, "cache": { "write": 0, "read": 8832 } }, "modelID": "gpt-6-astra", "providerID": "openai", "time": { "created": 1789261897916, "completed": 1789261920708 }, "finish": "tool-calls", "id": "msg_09851e0bc001w0CShElCI3htvh", "sessionID": "ses_f67ae44bbffehttioFiMPV7e59" }, "parts": [ { "type": "step-start", "id": "prt_09851e86b001PXmC2tSArV8mZp", "sessionID": "ses_f67ae44bbffehttioFiMPV7e59", "messageID": "msg_09851e0bc001w0CShElCI3htvh" }, { "type": "reasoning", "text": "**Verifying formula exhaustively**", "time": { "start": 1789261899885, "end": 1789261912524 }, "metadata": { "openai": { "itemId": "rs_0bd6d57de5365e0b016aa5f84be53487d28e6e0ce7253a86bc", "reasoningEncryptedContent": "gAAAAABqpfhYdOFNugkNMl-XrKE23yOxe8vXKWGASkMMLcy5N7XMt8UHCFWkdZ74kudyHMTe8sKMqS5i6wwfbxSvGqM7dU90E5sX8qjUYasRzHKgozE3oh76iAcPYc4NIJ9aFXLiCS4yZE8SmSVoHy2BgbUUMDiaF7mFo883IkZfbJO06ClXBWRFwItqLXyWQy40osLvtF2_GJTMkR84UU604jE8pttWw9yrX8LNZCnkz06NtNfqYhMZLfJlUjeZErwf72luUfGyJkRn5awlqUMMPnTlocww1NtuJDs-38ttbkv1zD84qptu3iZ-JfRC7dUNT8U_eHsp3uo11qmcGMzrWlsbAlbPJAISo--d6v4QtyLlrs1bpQ0QaHEWmSXsaYL1WApwxxQuyI8TuM-olvGFd5Ki27zfOaivakKdQw9qW06bqUFQ7e1X4u_-U4f6JgAtSFJo6YXCkwpMjPHqWYbSGeMXeBeYmIv1xy-vBgYOK2pcjPzfxmY3k4xjGb-JEG0CEN5DI67p8U2nuXqLCCLD79RiXcuTUZVmj1oY6M7-RmGJYFdUf10mnNsejxLBZb2YYiG3Li-zWsk_n47IU3UFAhHnT7nYFaTxkLa9JIys9pVF_6iB8tL2q43-B3CHPYAUPbJ2cTzpL861hHbeQrFORQEROsNctQ1l78A3TlcYTlBtNxJ1wR6bABG0Let8pOInueCx20fCvgi2YAnjlEqQ4HtamnkKnjwwRh3ZrjwTcYzGzQ0T7IiorfYEya3I4mzQbco26KoPYDLwkr420gNTyBz1shSTfsEWeNm4nFjFVfgxnAhi53ykHF3dHKhYb9wygzYXxKImrmfWXHKSv_MqRtHwSDkx9ee07eXEVa-fBOlzZt0T0He34YodT-k9tN6tvl4ibCM5521uKVtQ4IIjwwpHUTjPco4ZTpx8jo-Texecc7FsN7tgJIdIW4bNN-GQq2KMRm8GCJQD1oAYLYixyANtskXm1D8Yp6dZQLr7nG2HcLJzhDUzo5dWiJaEIWMXdGj9MXC67uLp9-ffulr9YmiFxn7Hu-8ckR4xUECodatbHeFPKMJiHjm_xRvJdtRsWmsXX3w3Lit2-Nuw53atw3hUg01mcVODMa4N1eXQYbFAGS4aZJyAo9xknUz5bnyL0_V1Oj298olfgxvoNDZbQDZasB2pP-eMJ-xejhj4jbd9P7zMszkxK4k3vDnedmy_Zt0UbmSOR1C5zOeqvvPH_GoSnuYY4tdMQz7LPnkCvrHozWfdkdRNclp31DMGwC-XT5_7WAJO8Q5K-hNdZRtviGrjKRU6GrsT8F_wyLvaAfQgXcJpTtbX1mIzWDaa9tN0UYSaVRrhOgaxmxAVSGmdY_DFvJIZPe1Oxdk7Vhvx8-Yw9E57vZUvF0v7MGzZeQkQPtuvUPMogWIdVkbRT_Pz_DUvn8yNCbkGFJI-1w51R_mpF2NkHJZj_ZGNlVB0-E2BaOZh7az6zg14zQCZds0oh_Q0Rdh99W0ezGnJeF3IKGzV_0kwiFum4AGn4wkTH6IaHxZyMPFMHZCIcuFvI7MQMzusjcx0xKAZdXTGq6HLPSnnKsmTZYLrV8NPrcg2Z82sFagT55XKVHeOiKxxOekkEk_xaX-ZpxEgfHWSDRqDlNinn7ONTJpY3mXqr5aZhUnvAqfxMwShTSKnYTGxA4ckdSfKCLR46-LPU2rLc47Xat81OYHtnBoIhvKBxEWz4AVbdJ4wSxYLz-qwi0zSQTm397Owz3o-qtJGOVmEZihpOIxwoiZvR6CbwwQRzJP0wO3ykgYAyKFC8fJgBZzJQ6T307mrsYNHObLCrVYyX3eE3MbyoHrTWiEJdcOcJ95-Cok6UlojhcoPE0Hk9cV1u3XhBXOJjGxFnN-bVktnll3Grf0yetlKBSW-TCklVK-UunSkpPNRGwSfWjs7bMv1DLf0Ou_iSwPW-YBBIS8R37zmFER-uJOivvdSjkJNKPCPI5GoJKwM3Z6ANn7t6_wcHceqisaD8_-UvtDemhvJvCESB4XGjdsPkyyNiBqVpv4IWLo9_zhwzVqLwdHQfEHzCMPwuotDyBMM43Tb3CMZIDjGEjhfJSjVWU3j--fuNnZg_RO8G5p4TgcdOGkgadxbR6rww44zU9Pr3vFFfs3fwsrZ8SBAn5fcLVS4mn9qQops_jkCJ-qKbrdsrfKjy6SwNgffjxDOGm0HzlAbJsC2mqh8eJBowIYBhec=" } }, "id": "prt_09851e86d001fQx7svUHhx8YIZ", "sessionID": "ses_f67ae44bbffehttioFiMPV7e59", "messageID": "msg_09851e0bc001w0CShElCI3htvh" }, { "type": "tool", "tool": "bash", "callID": "call_JziuMg3bIdajuaCbEmOBQ0MY", "state": { "status": "completed", "input": { "command": "python3 -c 'from itertools import combinations_with_replacement, product\ncases=0\nfor n in range(2,7):\n for a in combinations_with_replacement(range(1,7),n):\n for k in range(1,4):\n if any(a[j+1]-a[j]>k for j in range(n-1)): continue\n for i in range(n):\n ans=0\n if 0a[i-1]+(m-i)*k: lo=m+1\n else: hi=m\n c=lo-i-1\n ans=sum(a[i+1:lo])-(2*a[i-1]+(c+1)*k)*c//2\n b=a[:i]+a[i+1:]\n dp={v:b[0]-v for v in range(1,b[0]+1)}\n for x in b[1:]:\n dp={v:x-v+min((cost for u,cost in dp.items() if u<=v<=u+k),default=10**9) for v in range(1,x+1)}\n assert ans==min(dp.values()),(a,k,i,ans,dp)\n cases+=1\nprint(\"Verified\",cases,\"independent removals against exhaustive value DP.\")'" }, "output": "Verified 8778 independent removals against exhaustive value DP.\n", "metadata": { "output": "Verified 8778 independent removals against exhaustive value DP.\n", "exit": 0, "truncated": false }, "title": "python3 -c 'from itertools import combinations_with_replacement, product\ncases=0\nfor n in range(2,7):\n for a in combinations_with_replacement(range(1,7),n):\n for k in range(1,4):\n if any(a[j+1]-a[j]>k for j in range(n-1)): continue\n for i in range(n):\n ans=0\n if 0a[i-1]+(m-i)*k: lo=m+1\n else: hi=m\n c=lo-i-1\n ans=sum(a[i+1:lo])-(2*a[i-1]+(c+1)*k)*c//2\n b=a[:i]+a[i+1:]\n dp={v:b[0]-v for v in range(1,b[0]+1)}\n for x in b[1:]:\n dp={v:x-v+min((cost for u,cost in dp.items() if u<=v<=u+k),default=10**9) for v in range(1,x+1)}\n assert ans==min(dp.values()),(a,k,i,ans,dp)\n cases+=1\nprint(\"Verified\",cases,\"independent removals against exhaustive value DP.\")'", "time": { "start": 1789261920608, "end": 1789261920698 } }, "metadata": { "openai": { "itemId": "fc_0bd6d57de5365e0b016aa5f8586f0087d2ba1d381ba56630fa" } }, "id": "prt_0985219d3001o0VejVJmlgouVb", "sessionID": "ses_f67ae44bbffehttioFiMPV7e59", "messageID": "msg_09851e0bc001w0CShElCI3htvh" }, { "reason": "tool-calls", "type": "step-finish", "tokens": { "total": 22494, "input": 13134, "output": 343, "reasoning": 185, "cache": { "write": 0, "read": 8832 } }, "cost": 0, "id": "prt_0985239c1001JS4oI9UjPc5vli", "sessionID": "ses_f67ae44bbffehttioFiMPV7e59", "messageID": "msg_09851e0bc001w0CShElCI3htvh" } ] }, { "info": { "parentID": "msg_09851bb4e001vt60m8o1BKX8V3", "role": "assistant", "mode": "nudge-generation", "agent": "nudge-generation", "variant": "low", "path": { "cwd": "/private/var/folders/t7/b_w5m78j2mbb1ct051p9sv500000gn/T/nudge-opencode-generation-Al18sO", "root": "/" }, "cost": 0, "tokens": { "total": 24821, "input": 885, "output": 2304, "reasoning": 0, "cache": { "write": 0, "read": 21632 } }, "modelID": "gpt-6-astra", "providerID": "openai", "time": { "created": 1789261920710, "completed": 1789262014595 }, "finish": "tool-calls", "structured": { "status": "success", "reason": null, "hints": [ { "order": 1, "content": "Removing an endpoint always costs $0$. For an interior index $i$, only one new adjacent pair appears: $(a_{i-1},a_{i+1})$. Which condition can that pair violate?" }, { "order": 2, "content": "If the new gap is too large, decreasing its left endpoint only makes things worse. Keep the prefix unchanged and choose each remaining element, from left to right, as large as possible: $x_j=\\min(a_j,x_{\\text{previous}}+k)$. A decrease may force more decreases to its right." }, { "order": 3, "content": "Keep using the original indices after removing $i$. The upper bound propagated from $a_{i-1}$ to position $j>i$ is $a_{i-1}+(j-i)k$. Show that the optimal final value is $x_j=\\min(a_j,a_{i-1}+(j-i)k)$." }, { "order": 4, "content": "Position $j>i$ needs a decrease exactly when $a_j-jk>a_{i-1}-ik$. Since the original array is good, the sequence $a_j-jk$ is non-increasing. So the changed positions form one consecutive block starting at $i+1$, and you can binary-search its end." }, { "order": 5, "content": "Let $r$ be the first unchanged position after $i$, or $n+1$ if every remaining position changes. With $c=r-i-1$ and prefix sums $P$, the answer is\n$$P_{r-1}-P_i-c\\,a_{i-1}-k\\frac{c(c+1)}2.$$\nThe subtracted values form an arithmetic progression. Each deletion now needs one binary search and an $O(1)$ sum calculation. Use 64-bit arithmetic." } ], "editorial": "Deleting one element can trigger a whole chain of decreases. Fixing just the new gap is tempting, but the next gap may immediately break. The trick is to describe that entire chain without simulating it.\n\nAll indices below are **1-based original indices**; we do not renumber elements after deletion.\n\n### 1. Make every surviving element as large as possible\n\nRemoving the first or last element leaves a good array, so those answers are $0$.\n\nNow remove an interior element $a_i$. The only new adjacent pair is $(a_{i-1},a_{i+1})$. It is still sorted, but its gap may exceed $k$.\n\nKeep the prefix through $a_{i-1}$ unchanged. Then process the suffix from left to right, choosing\n$$x_j=\\min(a_j,x_{\\text{previous}}+k).$$\n\nThis takes the largest value permitted by both the original value and the previous chosen value.\n\nWhy is it still non-decreasing? The original value $a_j$ is at least the previous original value, which is at least $x_{\\text{previous}}$. Also, $x_{\\text{previous}}+k\\ge x_{\\text{previous}}$. Thus their minimum is at least $x_{\\text{previous}}$, and the gap is at most $k$.\n\nWhy is it optimal? Compare it with any feasible final array $y$. On the prefix, $y_j\\le a_j=x_j$, because operations only decrease values. Moving right, if $y_{\\text{previous}}\\le x_{\\text{previous}}$, then\n$$y_j\\le\\min(a_j,y_{\\text{previous}}+k)\n\\le\\min(a_j,x_{\\text{previous}}+k)=x_j.$$\nSo the greedy array is at least as large at every position as any feasible array. That minimizes the total number of decrements. Decreasing the prefix cannot buy us anything.\n\n### 2. Replace the simulation with a formula\n\nAfter removing $i$, position $j>i$ is exactly $j-i$ adjacency steps after position $i-1$. The bound propagated from that position is\n$$L_j=a_{i-1}+(j-i)k.$$\n\nThe greedy result is\n$$x_j=\\min(a_j,L_j).$$\n\nFor $j=i+1$, this is exactly the greedy rule. For a later position, suppose the formula holds for $j-1$. Then\n$$\\begin{aligned}\nx_j\n&=\\min(a_j,x_{j-1}+k)\\\\\n&=\\min(a_j,a_{j-1}+k,L_j)\\\\\n&=\\min(a_j,L_j),\n\\end{aligned}$$\nbecause the original array satisfies $a_j\\le a_{j-1}+k$.\n\nTherefore, the cost contributed by position $j$ is\n$$\\max\\bigl(0,a_j-a_{i-1}-(j-i)k\\bigr).$$\n\n### 3. Binary-search where the decreases stop\n\nDefine the transformed values\n$$d_j=a_j-jk.$$\nThe original gap bound gives\n$$d_{j+1}-d_j=(a_{j+1}-a_j)-k\\le0,$$\nso $d$ is non-increasing.\n\nPosition $j>i$ needs a decrease precisely when\n$$a_j>a_{i-1}+(j-i)k,$$\nor equivalently,\n$$d_j>a_{i-1}-ik.$$\n\nSince $d$ is non-increasing, this condition is true for an initial block of the suffix and false afterward. Once the chain stops, it stays stopped.\n\nBinary-search the first position $r\\in[i+1,n]$ satisfying\n$$a_r\\le a_{i-1}+(r-i)k.$$\nIf none exists, take $r=n+1$. The changed positions are exactly $i+1,\\ldots,r-1$.\n\nThere is no need to store $d$: the original inequality works directly in the binary search.\n\n### 4. Sum the block in constant time\n\nBuild prefix sums\n$$P_0=0,\\qquad P_j=\\sum_{p=1}^{j}a_p.$$\n\nLet $c=r-i-1$ be the number of changed elements. Their original sum is\n$$P_{r-1}-P_i.$$\nTheir final values form the arithmetic progression\n$$a_{i-1}+k,\\ a_{i-1}+2k,\\ \\ldots,\\ a_{i-1}+ck.$$\nThus\n$$\\boxed{\\text{answer}_i=P_{r-1}-P_i-c\\,a_{i-1}-k\\frac{c(c+1)}2.}$$\nIf $c=0$, the answer is simply $0$.\n\nFor example, take $a=[1,2,4,6]$ and $k=2$. Removing index $2$ leaves $[1,4,6]$. The suffix caps are $3$ and $5$, so the optimal array is $[1,3,5]$, costing $2$ operations. Fixing only the $4$ would leave a gap of $3$ afterward—the chain matters.\n\n### Correctness summary\n\nThe left-to-right greedy construction is good and componentwise maximizes the final array, so it minimizes operations. Its suffix values are exactly $\\min(a_j,a_{i-1}+(j-i)k)$. Monotonicity of $a_j-jk$ lets binary search identify exactly the positions that decrease. Prefix sums minus the arithmetic-progression sum then count exactly those decrements. Therefore every reported answer is optimal.\n\n### Complexity and implementation details\n\n- Prefix sums take $O(n)$ time.\n- Each interior deletion takes $O(\\log n)$ time.\n- Total: $O(n\\log n)$ time and $O(n)$ space per test case.\n\nUse `long long` for values, prefix sums, products, and answers. Binary-search caps can reach roughly $2\\cdot10^{14}$. In the code, the final-value sum is computed as `(first + last) * count / 2`. When the changed block is nonempty, both endpoint values are below their original values, hence at most $10^9$, so this product safely fits in 64 bits.\n\nThe searches always use the original array: the deletions are independent. For $n=2$, both answers are automatically $0$.", "solution": "#include \nusing namespace std;\n\nusing ll = long long;\n\nvoid setIO() {\n ios::sync_with_stdio(false);\n cin.tie(nullptr);\n}\n\nint main() {\n setIO();\n\n int t;\n cin >> t;\n while (t--) {\n int n;\n ll k;\n cin >> n >> k;\n\n vector a(n + 1), pref(n + 1), ans(n + 1);\n for (int i = 1; i <= n; ++i) {\n cin >> a[i];\n pref[i] = pref[i - 1] + a[i];\n }\n\n for (int i = 2; i < n; ++i) {\n int lo = i + 1, hi = n + 1;\n // Find the first suffix element that needs no decrease.\n while (lo < hi) {\n int mid = lo + (hi - lo) / 2;\n ll cap = a[i - 1] + (mid - i) * k;\n if (a[mid] > cap) lo = mid + 1;\n else hi = mid;\n }\n\n ll count = lo - i - 1;\n if (count == 0) continue;\n\n ll first = a[i - 1] + k;\n ll last = a[i - 1] + count * k;\n ll finalSum = (first + last) * count / 2;\n ans[i] = pref[lo - 1] - pref[i] - finalSum;\n }\n\n for (int i = 1; i <= n; ++i) {\n cout << ans[i] << (i == n ? '\\n' : ' ');\n }\n }\n}\n" }, "id": "msg_0985239c6001fzdVduoI8wYYYw", "sessionID": "ses_f67ae44bbffehttioFiMPV7e59" }, "parts": [ { "type": "step-start", "id": "prt_0985254c7001y64xrLjmJCfiID", "sessionID": "ses_f67ae44bbffehttioFiMPV7e59", "messageID": "msg_0985239c6001fzdVduoI8wYYYw" }, { "type": "tool", "tool": "StructuredOutput", "callID": "call_B9oYRkRJRvsRQi0lcsCGBCIW", "state": { "status": "completed", "input": { "status": "success", "reason": null, "hints": [ { "order": 1, "content": "Removing an endpoint always costs $0$. For an interior index $i$, only one new adjacent pair appears: $(a_{i-1},a_{i+1})$. Which condition can that pair violate?" }, { "order": 2, "content": "If the new gap is too large, decreasing its left endpoint only makes things worse. Keep the prefix unchanged and choose each remaining element, from left to right, as large as possible: $x_j=\\min(a_j,x_{\\text{previous}}+k)$. A decrease may force more decreases to its right." }, { "order": 3, "content": "Keep using the original indices after removing $i$. The upper bound propagated from $a_{i-1}$ to position $j>i$ is $a_{i-1}+(j-i)k$. Show that the optimal final value is $x_j=\\min(a_j,a_{i-1}+(j-i)k)$." }, { "order": 4, "content": "Position $j>i$ needs a decrease exactly when $a_j-jk>a_{i-1}-ik$. Since the original array is good, the sequence $a_j-jk$ is non-increasing. So the changed positions form one consecutive block starting at $i+1$, and you can binary-search its end." }, { "order": 5, "content": "Let $r$ be the first unchanged position after $i$, or $n+1$ if every remaining position changes. With $c=r-i-1$ and prefix sums $P$, the answer is\n$$P_{r-1}-P_i-c\\,a_{i-1}-k\\frac{c(c+1)}2.$$\nThe subtracted values form an arithmetic progression. Each deletion now needs one binary search and an $O(1)$ sum calculation. Use 64-bit arithmetic." } ], "editorial": "Deleting one element can trigger a whole chain of decreases. Fixing just the new gap is tempting, but the next gap may immediately break. The trick is to describe that entire chain without simulating it.\n\nAll indices below are **1-based original indices**; we do not renumber elements after deletion.\n\n### 1. Make every surviving element as large as possible\n\nRemoving the first or last element leaves a good array, so those answers are $0$.\n\nNow remove an interior element $a_i$. The only new adjacent pair is $(a_{i-1},a_{i+1})$. It is still sorted, but its gap may exceed $k$.\n\nKeep the prefix through $a_{i-1}$ unchanged. Then process the suffix from left to right, choosing\n$$x_j=\\min(a_j,x_{\\text{previous}}+k).$$\n\nThis takes the largest value permitted by both the original value and the previous chosen value.\n\nWhy is it still non-decreasing? The original value $a_j$ is at least the previous original value, which is at least $x_{\\text{previous}}$. Also, $x_{\\text{previous}}+k\\ge x_{\\text{previous}}$. Thus their minimum is at least $x_{\\text{previous}}$, and the gap is at most $k$.\n\nWhy is it optimal? Compare it with any feasible final array $y$. On the prefix, $y_j\\le a_j=x_j$, because operations only decrease values. Moving right, if $y_{\\text{previous}}\\le x_{\\text{previous}}$, then\n$$y_j\\le\\min(a_j,y_{\\text{previous}}+k)\n\\le\\min(a_j,x_{\\text{previous}}+k)=x_j.$$\nSo the greedy array is at least as large at every position as any feasible array. That minimizes the total number of decrements. Decreasing the prefix cannot buy us anything.\n\n### 2. Replace the simulation with a formula\n\nAfter removing $i$, position $j>i$ is exactly $j-i$ adjacency steps after position $i-1$. The bound propagated from that position is\n$$L_j=a_{i-1}+(j-i)k.$$\n\nThe greedy result is\n$$x_j=\\min(a_j,L_j).$$\n\nFor $j=i+1$, this is exactly the greedy rule. For a later position, suppose the formula holds for $j-1$. Then\n$$\\begin{aligned}\nx_j\n&=\\min(a_j,x_{j-1}+k)\\\\\n&=\\min(a_j,a_{j-1}+k,L_j)\\\\\n&=\\min(a_j,L_j),\n\\end{aligned}$$\nbecause the original array satisfies $a_j\\le a_{j-1}+k$.\n\nTherefore, the cost contributed by position $j$ is\n$$\\max\\bigl(0,a_j-a_{i-1}-(j-i)k\\bigr).$$\n\n### 3. Binary-search where the decreases stop\n\nDefine the transformed values\n$$d_j=a_j-jk.$$\nThe original gap bound gives\n$$d_{j+1}-d_j=(a_{j+1}-a_j)-k\\le0,$$\nso $d$ is non-increasing.\n\nPosition $j>i$ needs a decrease precisely when\n$$a_j>a_{i-1}+(j-i)k,$$\nor equivalently,\n$$d_j>a_{i-1}-ik.$$\n\nSince $d$ is non-increasing, this condition is true for an initial block of the suffix and false afterward. Once the chain stops, it stays stopped.\n\nBinary-search the first position $r\\in[i+1,n]$ satisfying\n$$a_r\\le a_{i-1}+(r-i)k.$$\nIf none exists, take $r=n+1$. The changed positions are exactly $i+1,\\ldots,r-1$.\n\nThere is no need to store $d$: the original inequality works directly in the binary search.\n\n### 4. Sum the block in constant time\n\nBuild prefix sums\n$$P_0=0,\\qquad P_j=\\sum_{p=1}^{j}a_p.$$\n\nLet $c=r-i-1$ be the number of changed elements. Their original sum is\n$$P_{r-1}-P_i.$$\nTheir final values form the arithmetic progression\n$$a_{i-1}+k,\\ a_{i-1}+2k,\\ \\ldots,\\ a_{i-1}+ck.$$\nThus\n$$\\boxed{\\text{answer}_i=P_{r-1}-P_i-c\\,a_{i-1}-k\\frac{c(c+1)}2.}$$\nIf $c=0$, the answer is simply $0$.\n\nFor example, take $a=[1,2,4,6]$ and $k=2$. Removing index $2$ leaves $[1,4,6]$. The suffix caps are $3$ and $5$, so the optimal array is $[1,3,5]$, costing $2$ operations. Fixing only the $4$ would leave a gap of $3$ afterward—the chain matters.\n\n### Correctness summary\n\nThe left-to-right greedy construction is good and componentwise maximizes the final array, so it minimizes operations. Its suffix values are exactly $\\min(a_j,a_{i-1}+(j-i)k)$. Monotonicity of $a_j-jk$ lets binary search identify exactly the positions that decrease. Prefix sums minus the arithmetic-progression sum then count exactly those decrements. Therefore every reported answer is optimal.\n\n### Complexity and implementation details\n\n- Prefix sums take $O(n)$ time.\n- Each interior deletion takes $O(\\log n)$ time.\n- Total: $O(n\\log n)$ time and $O(n)$ space per test case.\n\nUse `long long` for values, prefix sums, products, and answers. Binary-search caps can reach roughly $2\\cdot10^{14}$. In the code, the final-value sum is computed as `(first + last) * count / 2`. When the changed block is nonempty, both endpoint values are below their original values, hence at most $10^9$, so this product safely fits in 64 bits.\n\nThe searches always use the original array: the deletions are independent. For $n=2$, both answers are automatically $0$.", "solution": "#include \nusing namespace std;\n\nusing ll = long long;\n\nvoid setIO() {\n ios::sync_with_stdio(false);\n cin.tie(nullptr);\n}\n\nint main() {\n setIO();\n\n int t;\n cin >> t;\n while (t--) {\n int n;\n ll k;\n cin >> n >> k;\n\n vector a(n + 1), pref(n + 1), ans(n + 1);\n for (int i = 1; i <= n; ++i) {\n cin >> a[i];\n pref[i] = pref[i - 1] + a[i];\n }\n\n for (int i = 2; i < n; ++i) {\n int lo = i + 1, hi = n + 1;\n // Find the first suffix element that needs no decrease.\n while (lo < hi) {\n int mid = lo + (hi - lo) / 2;\n ll cap = a[i - 1] + (mid - i) * k;\n if (a[mid] > cap) lo = mid + 1;\n else hi = mid;\n }\n\n ll count = lo - i - 1;\n if (count == 0) continue;\n\n ll first = a[i - 1] + k;\n ll last = a[i - 1] + count * k;\n ll finalSum = (first + last) * count / 2;\n ans[i] = pref[lo - 1] - pref[i] - finalSum;\n }\n\n for (int i = 1; i <= n; ++i) {\n cout << ans[i] << (i == n ? '\\n' : ' ');\n }\n }\n}\n" }, "output": "Structured output captured successfully.", "metadata": { "valid": true }, "title": "Structured Output", "time": { "start": 1789262014508, "end": 1789262014510 } }, "metadata": { "openai": { "itemId": "fc_0d5f7d214f2fe320016aa5f867a2d087d1ae4445d790619fbe" } }, "id": "prt_0985254c8001pJFiPWMkHL88wY", "sessionID": "ses_f67ae44bbffehttioFiMPV7e59", "messageID": "msg_0985239c6001fzdVduoI8wYYYw" }, { "reason": "tool-calls", "type": "step-finish", "tokens": { "total": 24821, "input": 885, "output": 2304, "reasoning": 0, "cache": { "write": 0, "read": 21632 } }, "cost": 0, "id": "prt_09853a880001eIKBj49T41xiNN", "sessionID": "ses_f67ae44bbffehttioFiMPV7e59", "messageID": "msg_0985239c6001fzdVduoI8wYYYw" } ] } ] }