在算法竞赛中维护区间和是个很经典的问题。在数组中求区间 [l,r] 的和 sum[l,r] 显然可以用前缀和来优化。那么这个思想能不能推广呢比如现在有一个函数 f需要求 ∑ilrf(i)显然这个式子也可以转化为 sum[1,r]−sum[1,l−1]。为什么要做这步转化呢假设 f 是个周期函数那么直接求 sum[l,r] 可能会面临左右边界都不满一个周期的情况需要进行很多边界讨论。而显然此时 sum[1,i] 是更好求的因为我们只需要对右边界讨论。又比如 f 是个从 1 开始有规律的函数此时 sum[1,i] 显然也很好求。例题CF2036F题意给定 l,r,i,k求区间 [l,r] 上所有满足 x≢k(mod 2i) 的 x 的异或和。区间异或和显然也可以通过前缀异或和转化XOR[l,r]XOR[1,r]⊕XOR[1,l−1]。本题有个前置知识记 f(i) 为 从 1 到 i 的异或和则 f 以 4 为周期存在规律。https://m.douban.com/topic/480726565/https://www.douban.com/topic/480726537/?_spm_idNzk1MTIyMTghttps://m.douban.com/topic/480726537/https://www.douban.com/topic/480726505/?_spm_idNzk1MTIyMTghttps://m.douban.com/topic/480726505/https://www.douban.com/topic/480726477/?_spm_idNzk1MTIyMTghttps://m.douban.com/topic/480726477/https://www.douban.com/topic/480726446/?_spm_idNzk1MTIyMTghttps://m.douban.com/topic/480726446/https://www.douban.com/topic/480726419/?_spm_idNzk1MTIyMTghttps://m.douban.com/topic/480726419/https://www.douban.com/topic/480726389/?_spm_idNzk1MTIyMTghttps://m.douban.com/topic/480726389/https://www.douban.com/topic/480726358/?_spm_idNzk1MTIyMTghttps://m.douban.com/topic/480726358/https://www.douban.com/topic/480726330/?_spm_idNzk1MTIyMTghttps://m.douban.com/topic/480726330/https://www.douban.com/topic/480726304/?_spm_idNzk1MTIyMTghttps://m.douban.com/topic/480726304/https://www.douban.com/topic/480726274/?_spm_idNzk1MTIyMTghttps://m.douban.com/topic/480726274/https://www.douban.com/topic/480726243/?_spm_idNzk1MTIyMTghttps://m.douban.com/topic/480726243/https://www.douban.com/topic/480726207/?_spm_idNzk1MTIyMTghttps://m.douban.com/topic/480726207/https://www.douban.com/topic/480726177/?_spm_idNzk1MTIyMTghttps://m.douban.com/topic/480726177/https://www.douban.com/topic/480726148/?_spm_idNzk1MTIyMTghttps://m.douban.com/topic/480726148/https://www.douban.com/topic/480726116/?_spm_idNzk1MTIyMTghttps://m.douban.com/topic/480726116/https://www.douban.com/topic/480726087/?_spm_idNzk1MTIyMTghttps://m.douban.com/topic/480726087/https://www.douban.com/topic/480726058/?_spm_idNzk1MTIyMTghttps://m.douban.com/topic/480726058/https://www.douban.com/topic/480726026/?_spm_idNzk1MTIyMTghttps://m.douban.com/topic/480726026/https://www.douban.com/topic/480725994/?_spm_idNzk1MTIyMTghttps://m.douban.com/topic/480725994/https://www.douban.com/topic/480725963/?_spm_idNzk1MTIyMTghttps://m.douban.com/topic/480725963/https://www.douban.com/topic/480725929/?_spm_idNzk1MTIyMTghttps://m.douban.com/topic/480725929/https://www.douban.com/topic/480725895/?_spm_idNzk1MTIyMTghttps://m.douban.com/topic/480725895/https://www.douban.com/topic/480725861/?_spm_idNzk1MTIyMTghttps://m.douban.com/topic/480725861/https://www.douban.com/topic/480725828/?_spm_idNzk1MTIyMTghttps://m.douban.com/topic/480725828/https://www.douban.com/topic/480725794/?_spm_idNzk1MTIyMTghttps://m.douban.com/topic/480725794/https://www.douban.com/topic/480725761/?_spm_idNzk1MTIyMTghttps://m.douban.com/topic/480725761/https://www.douban.com/topic/480725730/?_spm_idNzk1MTIyMTghttps://m.douban.com/topic/480725730/https://www.douban.com/topic/480725698/?_spm_idNzk1MTIyMTghttps://m.douban.com/topic/480725698/https://www.douban.com/topic/480725666/?_spm_idNzk1MTIyMTghttps://m.douban.com/topic/480725666/https://www.douban.com/topic/480725635/?_spm_idNzk1MTIyMTghttps://m.douban.com/topic/480725635/https://www.douban.com/topic/480725605/?_spm_idNzk1MTIyMTghttps://m.douban.com/topic/480725605/https://www.douban.com/topic/480725573/?_spm_idNzk1MTIyMTghttps://m.douban.com/topic/480725573/https://www.douban.com/topic/480725539/?_spm_idNzk1MTIyMTghttps://m.douban.com/topic/480725539/https://www.douban.com/topic/480725507/?_spm_idNzk1MTIyMTghttps://m.douban.com/topic/480725507/https://www.douban.com/topic/480725465/?_spm_idNzk1MTIyMTghttps://m.douban.com/topic/480725465/https://www.douban.com/topic/480725426/?_spm_idNzk1MTIyMTghttps://m.douban.com/topic/480725426/https://www.douban.com/topic/480725387/?_spm_idNzk1MTIyMTghttps://m.douban.com/topic/480725387/https://www.douban.com/topic/480725341/?_spm_idNzk1MTIyMTghttps://m.douban.com/topic/480725341/https://www.douban.com/topic/480725298/?_spm_idNzk1MTIyMTghttps://m.douban.com/topic/480725298/https://www.douban.com/topic/480725254/?_spm_idNzk1MTIyMTghttps://m.douban.com/topic/480725254/https://www.douban.com/topic/480725213/?_spm_idNzk1MTIyMTghttps://m.douban.com/topic/480725213/https://www.douban.com/topic/480725172/?_spm_idNzk1MTIyMTghttps://m.douban.com/topic/480725172/https://www.douban.com/topic/480725131/?_spm_idNzk1MTIyMTghttps://m.douban.com/topic/480725131/https://www.douban.com/topic/480725086/?_spm_idNzk1MTIyMTghttps://m.douban.com/topic/480725086/https://www.douban.com/topic/480725047/?_spm_idNzk1MTIyMTghttps://m.douban.com/topic/480725047/https://www.douban.com/topic/480725008/?_spm_idNzk1MTIyMTghttps://m.douban.com/topic/480725008/https://www.douban.com/topic/480724964/?_spm_idNzk1MTIyMTghttps://m.douban.com/topic/480724964/https://www.douban.com/topic/480724926/?_spm_idNzk1MTIyMTghttps://m.douban.com/topic/480724926/https://www.douban.com/topic/480724881/?_spm_idNzk1MTIyMTghttps://m.douban.com/topic/480724881/https://www.douban.com/topic/480724841/?_spm_idNzk1MTIyMTghttps://m.douban.com/topic/480724841/https://www.douban.com/topic/480724798/?_spm_idNzk1MTIyMTghttps://m.douban.com/topic/480724798/https://www.douban.com/topic/480724755/?_spm_idNzk1MTIyMTghttps://m.douban.com/topic/480724755/https://www.dou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