(b) For every approximately concave functionf, cache sizek≥2, and page requestsequence that conforms tof, the page fault rate of the LRU policy is at mostαf(k)plus an additive term that goes to 0 with the sequence length.(c) There exists a choice of an approximately concave functionf, a cache sizek≥2,and an arbitrarily long page request sequence that conforms tof, such that thepage fault rate of the FIFO policy is bounded away fromαf(k).Parts (a) and (b) prove the worst-case optimality of the LRU policy in a strongand fine-grained sense,f-by-fandk-by-k. Part (c) differentiates LRU from FIFO,as the latter is suboptimal for some (in fact, many) choices offandk.The guarantees in Theorem 1.1 are so good that they are meaningful even whentaken at face value—for strongly sublinearf’s,αf(k) goes to 0 reasonably quicklywithk. The precise definition ofαf(k) fork≥2 isαf(k) =k-1f-1(k+ 1)-2,(1.1)where we abuse notation and interpretf-1(y) as the smallest value ofxsuch thatf(x) =y. That is,f-1(y) denotes the smallest window length in which page requestsforydistinct pages might appear. As expected, for the functionf(n) =nwe haveαf(k) = 1 for allk. (With no restriction on the input sequence, an adversary canforce a 100% fault rate.) Iff(n) =d√ne, however, thenαf(k) scales with 1/√k.Thus with a cache size of 10,000, the page fault rate is always at most 1%. Iff(n) =d1 + log2ne, thenαf(k) goes to 0 even faster withk, roughly ask/2k.1.3.2 Proof of Theorem 1.1This section proves the first two parts of Theorem 1.1; part (c) is left as Exercise 1.4.Part (a).To prove the lower bound in part (a), fix an approximately concave func-tionfand a cache sizek≥2. Fix a deterministic cache replacement policyA.We construct a page sequenceσthat uses onlyk+1 distinct pages, so at any giventime step there is exactly one page missing from the algorithm’s cache. (Assume thatthe algorithm begins with the firstkpages in its cache.) The sequence comprisesk-1 blocks, where thejth block consists ofmj+1consecutive requests for the samepagepj, wherepjis the unique page missing from the algorithmA’s cache at thestart of the block. (Recall thatmyis the number of values ofxsuch thatf(x) =y.)This sequence conforms tof(Exercise 1.3).By the choice of thepj’s,Aincurs a page fault on the first request of a block,