A cuckoo hashing variant with improved memory utilization and insertion time

A CUCKOO HASHING V ARIANT WITH IMPRO VED MEM OR Y UTILIZA TION AND INSER TION TIME EL Y PORA T AND BAR SHALEM 1. abstract Cuck o o hashing [4] is a multiple choice hashing scheme in which each item can b e placed in m ultiple lo catio ns, and collisions are r esolved b y moving items to their alternative loc a tions. In the cla ssical implementation o f tw o-way cuc koo ha s hing, the memory is partitioned int o contiguous disjoint fixed-size buckets. Each item is hashed to t wo buck e ts, and may b e stored in a ny o f the po sitions within those buc kets. Ref. [2] analyzed a v ariation in which the buck ets a re contiguous and ov erla p. How ever, many systems retrie ve da ta from seconda ry stor age in same-size blo cks called pages. F etching a pag e is a rela tively exp ensive pro cess; but once a pa ge is fetched, its conten ts c a n b e access ed order s of mag nitude faster. W e utilize this pr o p erty o f memory retriev al, presenting a v ariant of cuck o o hashing incorp o rating the following constraint: each buck et must be fully con ta ined in a sing le page, but buck ets are not necessarily co nt ig uous. Empirical results show that this mo dification increases memo ry utilization an d decrease s the n umber of itera tions required to insert an item. If each item is hashed to tw o buck ets of capa c ity tw o, the pa ge size is 8, a nd ea ch buck et is fully co ntained in a single page, the memory utilization equals 89.71% in the classical co nt ig uous disjoin t bucket v aria nt , 93.7 8% in the contiguous ov er lapping buck et v ar iant, and increases to 97.46 % in our new non-contiguous buc ket v ariant. When the memory utilization is 92% and we use breadth first sea rch to lo ok for a v acant position, the n umber of iter ations requir ed to insert a new item is dramatica lly reduced fr om 545 in the contiguous o verlapping buc kets v ariant to 52 in our new non-contiguous buck et v ariant. In addition to the empirica l results, we present a theoretical low er b ound on the memory utilization of our v ariation as a function o f the pag e siz e . 2. Introduction Cuck o o hashing [4] is a multiple choice has hing scheme in which each item can b e placed in m ultiple lo cations, and co llisions are r esolved by moving items to their alternative lo cations. This hashing scheme resembles the cuck o o ’s nes ting habits: the c uck o o lays its eg gs in other birds’ nests. When the cuc ko o chic k hatches, it pushes the other eggs o ut of the nest. Hence the name “cuck o o ha shing.” As Ref. [2] expla ins, analysis o f ha shing is similar to the analys is of balls and bins. Hashing an item to a memory lo cation corresp onds to throwing a ball in to a bin. Insights from ba lls and bins pro cesse s led to brea kthroughs in hashing metho ds. F or exa mple, if we throw n ba lls int o n bins indep endently and uniformly , it is highly probable tha t the largest bin will get (1 + o (1)) log ( n ) / loglog ( n ) balls. Azar et. a l [1 0] found that if each ball selects tw o bins independently and uniformly , and is pla ced in the bin with fewer balls, the final distribution is m uch mor e uniform. This led to hashing each item to one of tw o p os sible buck ets, decreasing the load on the most-loa ded buc ket to log ( log ( n )) + O (1) w ith high pro bability . In gener a l, if each item is ha shed into d ≥ 2 buck ets, the maximum load decrea ses to log ( log ( n )) / log ( d ) + O (1) . Cuck o o has hing [4] is an extension of tw o-wa y hashing . Each i tem is hashed to a few po ssible buc kets, and existing items may be mov ed to their alternate buck ets in order to free space for a new item. Ther e are many v ariants of cuck o o hashing. The g oals of cuck o o ha shing are to incr ease memory utilization (the num b e r o f items that can b e succe s sfully hashed to a given memory size) and to decre ase insertion co mplexit y . Pagh and Ro dler [4] ana lyzed hashing of e ach item to d = 2 1 A CUCKOO HASHING V ARIANT WITH IMPRO VED MEMOR Y UTILIZA TION AND INS ER TION TIME 2 buc kets of ca pacity k = 1 , and demonstra ted that moving items during inserts r esults in 50% space utilization with high probability . F o takis et. al. [5 ] ana lyzed hashing o f ea ch item into mo re than t wo buck ets. Ref. [6] analyzed a practically-imp or tant case in which each item is hashed to d = 2 buc kets of ca pacity k = 2 . Refs. [7, 8] found tight memor y utilization thres holds for d = 2 buck ets of any siz e k ≥ 2 . Specifically , they pr ov ed that the memory utilization fo r d = 2 and k = 2 is 89.7%. Ref. [1] prov ed that the ma ximu m memory utilization thresho lds for d ≥ 3 a nd k = 1 a re equal to the previo us ly known thresholds for the r andom k-XORSA T pr oblem. Ref. [12, 13] developed a tight formula for memory utilization for any d ≥ 3 and k = 1 and Ref. [11] extended the formula to a ny d ≥ 3 and k ≥ 1 . c omment adde d. While this work was being co mpleted, w e b ecame aware of Ref. [14] which prop osed a model wher e the memo ry is divided in to pages a nd each key has several po ssible lo cations on a single page a s well as additional c ho ices on a second ba ckup page. They pro v ide int e r esting exp erimental results. In a classical implementation of t wo-wa y cuckoo hashing, the memory is partitioned in to co n- tiguous disjoint fixed-sized buc kets o f size k . Each item is hashed to 2 buck e ts and may b e stored in a ny o f the 2 k lo cations within those buck ets. Ref. [2] analyze a v a riation in which the buc kets ov er lap. F o r exa mple, if the buck et capacity k is 3, the disjoint buck et memory lo ca - tions are: { 0 , 1 , 2 } , { 3 , 4 , 5 } , { 6 , 7 , 8 } , . . . . wher eas the ov erla pping buc ket memory loca tio ns are: { 0 , 1 , 2 } , { 1 , 2 , 3 } , { 3 , 4 , 5 } , . . . . Their empirical results show that this v ariation increa ses memory ut iliza tion fro m 89 .7% to 96 .5% for d = 2 and k = 2 . Ho wev er, many sy s tems retrieve data fr om secondary storage in s ame-size blo cks called pa ges. F etching a pag e is a relatively exp ensive pro c e ss, but once a pa ge is fetc hed, its conten ts ca n b e access ed orders of magnitude mor e quickly . W e utilize this prop erty of memory retriev al to pre s ent a v ariant of cuck o o hashing req uiring that each buck et b e fully contained in a single page but buck ets are not necessar ily contiguous. In this pa per we compare the following three v ariants of cuck o o hashing: (1) CUCKOO -CHOOSE-K- the algo rithm introduced in this pap er. The buck ets are any k cells in a page, not necessa rily in co nt ig uous lo c a tions. There ar e  t k  buc kets in a page, where t is the size of the page. (2) CUCKOO -OVERLAP [2]- The buc kets a re contiguous and ov e rlap. Here we assume that all buck ets are fully contained in a single pag e, so there are t − k + 1 buc kets in a page. This is a generaliza tion o f Ref. [2]. Orig ina lly Ref. [2] did not cons ider dividing the memo ry in to pa g es. (3) CUCKOO -DISJOINT [4]- The buck ets are contiguous a nd not ov erlapping . There ar e t/ k buc kets in a page. This is a g eneralization of Ref. [4]. Or ig inally Ref. [4] did not consider larger buck ets. Note that algo rithm CUCKO O -DISJOINT is the e x treme case of the CUCKOO-OVERLA P and CUCKOO-CHOO SE_K algorithms when the size of the pag e t equa ls the s ize o f the buck et k . W e pr ove theoretically a nd present empirical evidence that our CUCKO O-CHOOSE-K mo d- ification increa ses memor y utilization. Moreov er , using the classical cuck o o hashing s cheme, an item inser tion requires multiple loo k-ups of candidates to displace. Empirical results show that our mo dification dramatically decreases the num ber of candidat e lo ok-ups required to insert an item compared to Ref. [2]. In the ov er lapping buck ets v aria nt [2], some buck ets are split betw een t wo pages, so that each item r esides in up to 2 d pages . In our v ariant, each buck et is fully contained in a single page. An app ealing e xp e r iment al result is that CUCKOO-CHOOSE-K memory utilization conv erg e s very quickly a s a function of the pag e s ize t . When k = 2 and t = 16 , memo r y utilization is 0.9763 . This v alue is almost identical to memory utilization when t = 2 20 which equals 0.976 7. A CUCKOO HASHING V ARIANT WITH IMPRO VED MEMOR Y UTILIZA TION AND INS ER TION TIME 3 Sym bo l Description Comments n n um b er o f vertices n → ∞ (hash table capacity) m n um b er o f edges m ≤ n , m → ∞ (hashed items) d n um b er o f buck ets each t y pica l v alue: 2, dk > 2 item is hashed to k buc ket size t y pica l v alue: 2-3, dk > 2 t page size t ≥ k . k divides t . t divides n . g n um b er o f pages g = n/t . β memory utilization β = m n , 0 < β ≤ 1 V a set o f vertices | V | = v S = S (V,E) a sub-graph v v = | V | dk ≤ v < m . (if v ≥ m or v < dk then S cannot fail). x x = v n dk n ≤ x < β . ǫ a small constant 0 < ǫ ≪ 1 δ a small constant 0 < δ ≪ 1 T able 1. Parameter names and descr iptions CUCKOO-OVERLAP memory utilization when t = 16 is 0 .9494, and CUCKOO-DISJO INT mem- ory utilization is only 0.8 970. If w e allow a tiny gap of one inside the buck e ts ( t = 3) , memory utilization increa ses from 0.9229 (CUCKOO -OVERLAP ) to 0.948 0 (CUCKOO- CHO OSE-K). T able 4.2 sp ecifies the parameter s used in our ana ly sis. 3. Theoretical anal y sis W e ca n determine the succes s o f cuckoo hashing b y analyzing t he cuckoo hyper g raph. The vertices of the g r aph are the memory lo cations. The hyper-edges of the gr a ph co nnect all the memory lo cations where each item co uld b e placed. Reca ll that each item can b e placed in d buc kets chosen uniformly and indep endently o f other items. Each bucket is comp osed of a ny k lo cations in a page. It is well known (see, e.g ., ref. [2] fo r a pro of ) that a cuck o o hash fails if and only if there is a sub-graph S with v vertices and mo re than v edges. W e say that a sub-gr aph S has faile d if it has more edges than vertices. W e will b egin by analyzing the probability o f succes s of CUCKO O -CHOOSE-K for the case where the page siz e t eq uals the array size n . This simple and sp ecial ca se is presented here to in tr o duce the main ideas applied in the following sec tio n, where we a nalyze the g eneral case where the page size is a finite cons ta nt (indep endent of n ). 3.1. Memo ry util ization when the page size t equals the arr ay size n . An ana lysis of memory utilization has be en p erformed previous ly in [5]. In their analysis, they assume k = 1 and prov e that if d ≥ 2 β log  e β 1 − β  , then the ha s hing will b e succe s sful with a pro bability o f at lea st 1 − O ( n 4 − 2 d ) . Here w e der ive a similar co nstraint on memory utilization β . W e so lve the cons traint n umerically for differen t v alues of k a nd d , and obtain a low er bo und on p oss ible memory util ization for the sp ecified v alues. W e per form the analy s is us ing a mo dification of the metho d in [5], which we will later generalize to page sizes b eing equa l to any g iven co nstant. W e will b o und the failure probability using the union b ound. But first, we would like to reduce redundant summations. W e o bserve that: A CUCKOO HASHING V ARIANT WITH IMPRO VED MEMOR Y UTILIZA TION AND INS ER TION TIME 4 • If there exists a sub-graph S ( V , E ) with | E | > | V | and there exists an edge that has exac tly one vertex v 0 outside o f V, then the sub-gr aph S ′ ( V ′ = V ∪ { v 0 } , E ′ ) als o ha s more edge s than vertices since | E ′ | ≥ | E | + 1 > | V | + 1 = | V ′ | . • If there exists a sub-gr aph S ( V , E ) , such that | V | = v and | E | > v + 1 then there exists a sub-graph S ′ ( V ′ , E ′ ) s uch that | E ′ | = | V ′ | + 1 . W e c an find such a sub-graph simply by adding vertices to V one by one un til we g e t a sub-gra ph where the num b er of edges equals the num ber o f vertices plus one. F or each sub-gr a ph S we define an indica tor v ariable Z S . Z S =      1 S ( V , E ) has | V | = v v ertices and | E | = v + 1 edg es AND There is no edge that connects V to exactly o ne v er tex from outside of V 0 other w is e If Z s = 1 , then we will say that S is a b ad sub-gr aph, and otherwise we will say that S is a go o d sub-graph. If the sum ov e r Z s of all sub-gra phs is equal to zero then every sub-graph is go o d then the cuck o o hash suc c e e de d . W e will find the the memory utilization β suc h that the sum ov e r Z s of all s ub-graphs is o (1) as n → ∞ . Let p hit be the probability that a r a ndom edge hits V . Let p 1 be the pro bability that a r andom edge co nnects V to ex actly one vertex from outside o f V and let p bad ( v ) be pro bability that a given sub-graph S ( V , E ) is bad. Lemma 1. p bad ( v ) =  m v +1  p v +1 hit (1 − p 1 − p hit ) m − ( v +1) Pr o of. Immediate fro m the definition of Z s . F or S to b e bad, exactly v + 1 edges out of m edg es m us t hit V and all the rest must miss V and m ust not c o nnect V to exactly one vertex from outside of V.  Lemma 2. The pr ob ability that a r andom e dge hits V is p hit ( v ) =  ( v k ) ( n k )  d Pr o of. Each item is hashed independently to d buc kets and the size of eac h buc ket is k . T he num ber of buck ets in V is therefore  v k  and the total num b er of buck ets is  n k  .  Lemma 3 . The pr ob abili t y that a r andom e dge c onne cts V to exactly one vertex fr om outside of V is p 1 = d ( n − v ) ( v k − 1 ) ( n k )  ( v k ) ( n k )  d − 1 Pr o of. d − 1 buck ets must a ll fall in V and one buck et must contain an y k − 1 vertices from V and any of the n − v vertices from outside of V .  Let P bad ( v ) b e the probability that there exists a sub-gr aph S with v vertices suc h that S is bad. A cc ording to the union b ound, P bad ( v ) < N ( v ) · p bad ( v ) , where N v =  n v  is the num b er of sub- graphs with v vertices. W e are go ing to ana lyze P bad ( v ) as n → ∞ . If for a ll v , P bad ( v ) = o  1 n  , then P n v = d k P bad ( v ) ≤ o (1) and the cuc koo hash s ucceeds with high probability . The analysis is similar to the ana lysis given in [5]. Let x 0 = exp  − 2 dk − 2  . W e divide the analys is in to tw o se c tions. In section 3.1.1 w e show that fo r any memory utilization β and ∀ dk n ≤ x < x 0 , P bad ( x ) is o  1 n  . In section 3 .1.2w e find the maximum memory utilization β such that ∀ x 0 ≤ x < β , P bad ( x ) is exp onentially small. 3.1.1. P bad ( x ) Analysi s for dk n ≤ x < x 0 . In this section we show that if dk > 2 then for any memory utilization β and ∀ dk n ≤ x < x 0 , P bad ( x ) is o  1 n  . Lemma 4. P bad ( x ) < c 0 ( x ) · c n 1 ( x ) , wher e c 0 ( x ) = e · x dk − 1 and c 1 ( x ) = e 2 x · x ( dk − 2) x . Pr o of. See app endix A.  A CUCKOO HASHING V ARIANT WITH IMPRO VED MEMOR Y UTILIZA TION AND INS ER TION TIME 5 Theorem 5. L et δ b e a smal l c onstant, 0 < δ ≪ 1. If dk ≥ 3 t hen for any lo ad β : 1. If dk n ≤ x ≤ dk n + δ , then P bad ( x ) < O  n − ( ( dk ) 2 − dk − 1 )  = o  1 n  . 2. If dk n + δ ≤ x ≤ x 0 − δ , then P bad ( x ) de cr e ases exp onential ly as n → ∞ . Pr o of. F or any memory utilization β , P bad ( x ) < c 0 ( x ) · c n 1 ( x ) , where c 0 ( x ) = e · x dk − 1 and c 1 ( x ) = e 2 x · x ( dk − 2) x . Note that c 0 ( x ) a nd c 1 ( x ) a r e indep endent of β . Recall that x 0 = exp  − 2 dk − 2  . ∀ dk n + δ ≤ x ≤ x 0 − δ , there exist a co nstant ǫ s uch that c 1 ( x ) < 1 − ǫ . W e obtain that: 1. If x → dk n , then P bad ( x ) < lim x → dk n c 0 ( x ) · c n 1 ( x ) = O  n − ( ( dk ) 2 − dk − 1 )  ≤ O  n − 5  = o  1 n  . 2. If dk n + δ ≤ x ≤ x 0 − δ, then c 1 ( x ) < 1 − ǫ , and P bad ( x ) < c 0 ( x ) · c n 1 ( x ) decreases exp onentially as n → ∞ .  3.1.2. P bad ( x ) Analy sis for x 0 ≤ x < β . In this section w e ar e going to find the maximum mem- ory utilization β such that ∀ x 0 ≤ x < β , the pro bability tha t there exists a ba d sub-graph is exp onentially small. Lemma 6. ∀ x 0 ≤ x < β , P bad ( x ) < O (1) · c n 5 ( x, β ) wher e c 5 ( x, β ) =  1 1 − x  (1 − x )  1 x  x  β β − x  ( β − x )  β x  x x dkx  1 − dk (1 − x ) x dk − 1 − x dk  β − x Pr o of. See app endix B  Theorem 7. If c 5 ( x, β ) < 1 − ǫ for al l x 0 ≤ x < β , then P bad ( x ) de cr e ases exp onential ly as n → ∞ . A ny memory utilization β t hat satisfies the c onstr aint is a lower b oun d on the p ossible memory u tilization. Pr o of. The theorem follows directly fro m the inequa lity P bad ( x ) < O (1) · c n 5 ( x, β ) .  Numerical so lutions to the constra int c 5 ( x, β ) < 1 indicate that the memo ry utiliza tio n of the CUCKOO-K a lgorithm is β C hoose − 2 ( k = 2 , d = 2 , t = n ) > 0 . 9 37 and β C hoose − 3 ( k = 3 , d = 2 , t = n ) > 0 . 993 . Our empirical r esults show that β C hoose − 2 ( k = 2 , d = 2 , t = n ) = 0 . 976 8 and β C hoose − 3 ( k = 3 , d = 2 , t = n ) = 0 . 9974 . The memory utilization for k d > 6 ra pidly ap- proaches one. Theoretical a nalysis p er formed by [8, 7 ] provided tight thresholds of the memory utilization of the CUCKOO-DISJO INT algo rithm. β Disj oint ( k = 2 , d = 2 , t = n ) = 0 . 8 970 and β Disj oint ( k = 3 , d = 2 , t = n ) = 0 . 9592 . Ref. [2] do not provide a theoretical memory utilization threshold for small k . T he empirica l results o f Ref. [2] show tha t in the CUCKOO-OVERLAP algorithm β Ov er lap ( k = 2 , d = 2 , t = n ) = 0 . 965 0 and β Ov er lap ( k = 3 , d = 2 , t = n ) = 0 . 994 5 . The theoretical a na lysis of the CUCKO O-DISJOINT algor ithm p erformed in [12, 1, 13] do es not apply for k > 1 and the theoretical analysis of the CUCKOO - DISJOINT algor ithm perfor med by [11] do es not apply for d < 3 . 3.2. Memo ry util ization when the page size t i s a giv en constan t. In this section we analyze the pr obability that the hashing fails for the case w here the page size t equals a constant. Let P f a il ( v ) b e the pro bability that there exists a sub-g raph S with v vertices suc h that S has more edges than vertices and every vertex is in at least one edge. Let x 1 = exp  − ( k +1) dk − ( k +1)  . Here aga in we div ide the analys is into t wo sections. In section 3.2.1 we show that for a ny memor y utilization β a nd ∀ dk n ≤ x < x 1 , P f a il ( x ) is o  1 n  . In section 3.2.2 we find the maximum memory utilization β such that ∀ x 1 ≤ x < β , P bad ( x ) is expo nent ia lly sma ll. 3.2.1. P f a il ( x ) Analysi s for dk n ≤ x < x 1 . In this section we show that for a ny memor y utilizatio n β and ∀ dk n ≤ x < x 1 , P f a il ( x ) is o  1 n  . The ana ly sis her e is similar to the case ab ove where t = n , how ever now w e need to take into consideration the distribution of the vertices ov er the pages . A CUCKOO HASHING V ARIANT WITH IMPRO VED MEMOR Y UTILIZA TION AND INS ER TION TIME 6 Let V be a given set o f vertices, and let v i be the num b er of vertices in pag e i . The probability that a random edge hits V is equal to p hit ( V , t ) =  P g i =1 ( v i k ) g · ( t k )  d ≤  1 g P g i =1  v i t  k  d . Let e p hit ( V , t ) =  1 g P g i =1  v i t  k  d be an upper b ound on p hit ( V , t ) . W e are go ing to use the following lemma which s tates that incr e a sing the page size re duces e p hit . Lemma 8. F or any set of vertic es V and any inte ger c, e p hit ( V , t ) ≥ e p hit ( V , ct ) . Pr o of. Since the function f ( v ) =  v t  k is conv ex, for any sequence of c pages , 1 c   v 1 t  k + . . . +  v c t  k  ≥  v 1 + ... + v c ct  k , thus multiplying the page size b y a factor c does not increase e p hit . F or convenience, we res trict our analysis to pa ges of size t = k · c , where c is any integer. The worst case is o btained when t = k which is equiv alent to the cla ssical CUCK OO-DISJOINT ha shing. W e will now analyze P f a il ( x ) and P bad ( x ) a s n → ∞ . Recall that β = m n is the memor y utilization, 0 < β ≤ 1 and x = v n .  Lemma 9. P f a il ( x ) < c 6 ( x ) · c n 7 ( x ) wher e c 6 ( x ) = e x d − 1 and c 7 ( x ) = e ( k +1) x k x ( dk − 1 − k ) x k . Pr o of. See app endix C.  Theorem 10. L et δ b e a smal l c onst ant , 0 < δ ≪ 1. If ( d − 1) dk ≥ 3 , then for any lo ad β : 1. If dk n ≤ x ≤ dk n + δ , then P f a il ( x ) < O  n − (( d − 1) dk − 1)  = o  1 n  . 2. If dk n + δ ≤ x ≤ x 1 − δ , then P f a il ( x ) de cr e ases exp onential ly as n → ∞ . Pr o of. F or any memory utilization β , P f a il ( x ) < c 6 ( x ) · c n 7 ( x ) where c 6 ( x ) = e x d − 1 and c 7 ( x ) = e ( k +1) x k x ( dk − 1 − k ) x k . Note that c 6 ( x ) and c 7 ( x ) are indep endent of β . Reca ll that x 1 = e xp  − ( k +1) dk − ( k +1)  . ∀ dk n + δ ≤ x ≤ x 1 − δ , there exist a co nstant ǫ s uch that c 7 ( x ) < 1 − ǫ . W e obtain that: 1. If x → dk n , then P f a il ( x ) < lim x → dk n c 6 ( x ) · c n 7 ( x ) = O  n − (( d − 1) dk − 1)  ≤ O  n − 2  = o  1 n  . 2. If dk n + δ ≤ x < x 1 − δ , then c 7 ( x ) < 1 − ǫ , and P f a il ( x ) < c 6 ( x ) · c n 7 ( x ) decreases exponentially as n → ∞ .  3.2.2. P bad ( x ) A nalysis for x 1 ≤ x < β . In this section we find the maximum memor y utilization β suc h that ∀ x 1 < x < β , the pr o bability that there exists a bad sub-graph is exp onentially small. W e examine the set of sub-graphs that hav e a g iven distribution a of v er tices over the pages. a = ( a 0 , ..., a t ) , where a i is the num b er of pag es that hav e i vertices. F or example, when the pag e size t was equal to n a nd the num b er of pages g was 1 , the num b er o f sub-graphs with v vertices was  n v  and the c o rresp onding a of those sub-g raphs was { 0 , ..., 0 , a v = 1 , 0 , ..., 0 } . by definition of a, (3.1) t X i =0 a i = g Lemma 11. When the p age size t is a given c onstant, t he numb er of differ ent p ossible values of a is p olynomia l in n . Pr o of. W e denote by # a The num b er of different po ssible v alues o f a. # a < g t =  n t  t . Since t is constant, # a is p olynomial in n .  Let P bad ( a ) be the probability that there exists a bad sub-g raph with dis tr ibution a of vertices ov er the pages. If P bad ( a ) is expo nenti ally small for every a, then the union b ound o ver a polynomial n um b er of all p ossible v alues of a is also exp onentially small. Let p bad ( a ) b e the probability that a given sub-gr aph S ( V , E ) with a = ( a 0 , ..., a t ) is bad. A CUCKOO HASHING V ARIANT WITH IMPRO VED MEMOR Y UTILIZA TION AND INS ER TION TIME 7 Let ˆ a = a g be a unit vector. When the page size t is a constant, the proba bility that a random edge hits V is: (3.2) p hit ( a ) = P t i = k a i  i k  g  t k  ! d = P t i = k ˆ a i  i k   t k  ! d and the pr obability that a random edge connects V to exactly one vertex from outside of V is (3.3) p 1 ( a ) = d P t i = k a i ( t − i )  i k − 1  g  t k  ! P t i = k a i  i k  g  t k  ! d − 1 = d P t i = k ˆ a i ( t − i )  i k − 1   t k  ! P t i = k ˆ a i  i k   t k  ! d − 1 Using the union b ound we obtain that (3.4) P bad ( a ) < N ( a ) · p bad ( a ) where N ( a ) is the num b er of sub-g raphs with a = ( a 0 , ..., a t ) . (3.5) N ( a ) =  g a 0 , ..., a t  t Y i =0  t i  a i F or the asymptotic b ehavior of N ( a ) , we are going to use the following lemma: Lemma 12.  g a 0 ,...,a t  ≤ Q t i =0  g a i  a i Pr o of. See App endix D.  Lemma 13. P bad (ˆ a, β ) < O (1 ) · c 8 (ˆ a, β ) · c n 9 (ˆ a, β ) wher e c 8 (ˆ a, β ) = p hit (1 − p 1 − p hit ) − 1  β − x x  and c 9 (ˆ a, β ) = Q t i =0  1 ˆ a i  ˆ a i t Q t i =0  t i  ˆ a i t  β β − x  ( β − x )  β x  x p x hit (1 − p 1 − p hit ) β − x Pr o of. See app endix E.  Theorem 14 . If c 9 (ˆ a, β ) < 1 − ǫ for al l ∀ x 1 ≤ x < β , then P bad (ˆ a, β ) de cr e ases exp onential ly as n → ∞ . Any memory u t ilization β that satisfies the c onst r aint is a lower b oun d on the p ossible memory u tilization. Pr o of. The theorem follows directly fro m the inequa lity P bad (ˆ a, β ) < O (1 ) · c 8 (ˆ a, β ) · c n 9 (ˆ a, β ) .  Theoretical low er bo unds of the memory utilization obtained from numerical solutions of the constraint c 9 (ˆ a, β ) < 1 are displayed in figure 4.1. 4. Empirical Resul ts The exp eriments w er e conducted with a similar proto co l to the one describ ed in [2]. In a ll exp eriments the n umber of the buck ets, d , was tw o . The capacity of each bucket k was either tw o or three. The size of the hash tables n w as 1 , 209 , 600 . The r ep o rted memory utilizatio n β is the mean memory utilization ov er tw ent y tria ls. The random hash functions were based on the Matlab “rand” function with the t wister method. Items w ere inser ted into the hash table one-b y-one un til an item co uld not b e inserted. The r esults o f b o th CUCK OO-CHOOSE-K and CUCKOO-OVERLAP were notably stable. In each ca se, the standard deviation w a s a few hundredths o f a p erce nt, so error bars would b e invisible in the fig ure. Such strongly pr e dictable be havior is appea ling from a practical standpo int . Since we a dded a paging cons tr aint, o ur results are not co mpa rable to previous works that do no t include a paging constraint. A CUCKOO HASHING V ARIANT WITH IMPRO VED MEMOR Y UTILIZA TION AND INS ER TION TIME 8 Figure 4.1. Memory utilization vs. page size. Empirical CUCK O O-CHOOSE-K (gr e e n), Empirical CUCKOO-OVERLAP (red), approximation formula of Empirica l CUCKOO-CHOO SE-K (blue), and theo r etical low er b o und of C UCK OO-CHOOSE-K (black). left: k = 2 , right: k = 3 . Figure 4.2. Num b er of lo okups required to insert a n item vs. memory utilization (left) and vs. page size (right). In the left figure the pa ge size is 8 . In the right figure the memor y utilization is 92 %. The left fig ure was smo othed with an av er aging filter. The v ar ia nce in the num b er of lo okups r equired to inser t a n item was m uch smallar in CUCKOO- C HO OSE-K. Exper imen ts show that CUCKOO-CHOOSE-K impro ves memory utilization sig nificantly even for a small page size t a nd a small buck et c a pacity k when co mpared to the classical cuck o o hash- ing CUCKO O-DISJOINT. It outper forms CUCKOO -OVERLAP as w ell. Recall that CUCKOO- DISJOINT is the extreme case of CUCKOO-CHOOSE-K and CUCKOO-OVERLAP when the page size is equal to buck et size k . The memor y utilization β C hoose − k conv erge s v er y quickly to its ma x im um v alue. F or example β C hoose − 2 ( t = 1 6) = 0 . 976 3 is almos t equal to β C hoose − 2 ( t = 1 , 20 9 , 60 0) = 0 . 9 767 . Whereas β Ov er lap − 2 ( t = 16) = 0 . 9 494 and β Disj oint − 2 = 0 . 8970 . A CUCKOO HASHING V ARIANT WITH IMPRO VED MEMOR Y UTILIZA TION AND INS ER TION TIME 9 The empirica l memory utilization can b e a pproximated very accura tely b y the following form ula s: (4.1) β C hoose − 2 ( t ) ≈ 0 . 977 ·  1 − ( t 0 . 764 ) − 2 . 604  (4.2) β C hoose − 3 ( t ) ≈ 0 . 997 ·  1 − ( t 1 . 011 ) − 2 . 998  The max im um approximation er ror is 0.0011 for k = 2 and 0 .0015 for k = 3 . The empirical res ults and their approximations are displayed in figure 4.1 together with empirica l results o f CUCKOO -OVERLAP . The memory utilization fo r k d > 6 rapidly appr oaches one (not display ed). CUCKOO-CHOO SE-K outp erfor ms CUCKOO-OVERLAP no t only in memory utilization, but in the num b er of iterations r equired to insert a new item as well. Figure 4.2 illustr ates the num b er of iter ations required to insert a new item when the has h ta bl e is 9 2% full and we use breadth first search to search fo r a v acant p osition. # itr C hoose − 2 ( t = 8 ) = 52, whereas # i tr Ov er lap − 2 ( t = 8) = 5 45 . Note that in these simulations we did not limit the n umber o f insert itera tions and we contin ued to insert items a s long as w e could find free loca tio ns. Mos t applications limit the n umber of inserted itera tions, and ma int a in a low memory utilization in or der to find a free lo cation easily . If an empty p osition is not found within a fixed num b er of itera tions, a rehas h is pe r formed or the item is placed outside of the cuck o o arr ay . Appendix A. Proof of Lemma 4 Here w e prov e that P bad ( x ) < c 0 ( x ) · ( c 1 ( x )) n , where c 0 ( x ) = e · x dk − 1 and c 1 ( x ) = e 2 x · x ( dk − 2) x . Pr o of. Recall that β = m n is the memo ry utilization, 0 < β ≤ 1 and x = v n , dk n ≤ x < β . (if v ≥ m or v < dk , then the s ub- g raph S c a nnot fa il). P bad ( v ) <  n v  · p bad ( v ) where p bad ( v ) =  m v +1  p v +1 hit (1 − p 1 − p hit ) m − ( v +1) , p hit ( v ) =  ( v k ) ( n k )  d , and p 1 = d ( n − v ) ( v k − 1 ) ( n k )  ( v k ) ( n k )  d − 1 .  n v  , p bad ,  m v +1  and p hit are b ounded by: (A.1)  n v  <  e · n v  v =  e x  xn . (A.2) p bad ( v ) =  m v + 1  p v +1 hit (1 − p 1 − p hit ) m − ( v +1) <  m v + 1  p v +1 hit . (A.3)  m v + 1  <  e · m v + 1  v +1 <  e · m v  v +1 =  e · β x  xn +1 <  e x  xn +1 . (A.4) p hit <  v n  dk = x dk . And we get that (A.5) P bad ( v ) <  n v  · p bad ( v ) <  e x  xn  e x  xn +1  x dk  xn +1 = c 0 ( x ) · c n 1 ( x ) . A CUCKOO HASHING V ARIANT WITH IMPRO VED MEMOR Y UTILIZA TION AND INS ER TION TIME 10  Appendix B. Proof Of Lemma 6 Here we pr ov e ∀ x 0 ≤ x < β , P bad ( x, β ) < O (1) · c n 5 ( x, β ) where c 5 ( x, β ) =  1 1 − x  (1 − x )  1 x  x  β β − x  ( β − x )  β x  x x dkx  1 − dk (1 − x ) x dk − 1 − x dk  β − x . Pr o of. Recall that: (B.1) P bad ( v ) < N ( v ) · p bad ( v ) =  n v  m v + 1  p v +1 hit (1 − p 1 − p hit ) m − ( v +1) .  n v  , p hit and p 1 are b ounded b y: (B.2)  n v  <  n n − v  n − v  n v  v =  1 1 − x  (1 − x ) n  1 x  xn , (B.3)  x − k n  dk < p hit < x dk , (B.4) dk (1 − x ) x  x − k n  dk < p 1 . Since x ≥ x 0 ≫ 1 n , k n , we neglect the term 1 n ≪ x in the following a pproximation of  m v +1  , and we neglect the term k n ≪ x in the low er b ounds for p hit and p 1 . As n → ∞ , these ter ms contribute to P bad ( v ) factors which are b ounded by O (1) . (B.5)  m v + 1  <  m m − ( v + 1)  m − ( v +1)  m v + 1  v +1 < O (1) ·  β β − x  ( β − x ) n − 1  β x  xn +1 (B.6) (1 − p 1 − p hit ) m − ( v +1) < O (1) ·  1 − dk (1 − x ) x dk − 1 − x dk  β n − ( xn +1) The pro of for the left inequalities is given in [5] and is also a sp ecial case of the more general inequality we prov e la ter in lemma 1 2. W e get that: (B.7) P bad ( v ) <  n v  · p bad ( v ) < O (1) · c 4 ( x, β ) · c n 5 ( x, β ) , where (B.8) c 4 ( x, β ) = ( β − x ) x dk − 1  1 − dk (1 − x ) x dk − 1 − x dk  − 1 and (B.9) c 5 ( x, β ) =  1 1 − x  (1 − x )  1 x  x  β β − x  ( β − x )  β x  x x dkx  1 − dk (1 − x ) x dk − 1 − x dk  β − x Since c 4 < O (1) , we g et: P bad ( x ) < O (1) · c n 5 ( x, β ) .  A CUCKOO HASHING V ARIANT WITH IMPRO VED MEMOR Y UTILIZA TION AND INS ER TION TIME 11 Appendix C. Proof of Lemma 9 Here we pr ov e P f a il ( v ) < c 6 ( x ) · c n 7 ( x ) where c 6 ( x ) = e x d − 1 and c 7 ( x ) = e ( k +1) x k x ( dk − 1 − k ) x k . Pr o of. According to lemma 8, for any set of vertices V, e p hit decreases when the page size t is m ultiplied by an in teger . F or simplicity , we res trict our a nalysis to pa ges of s ize t = k · c , where c is any integer. The worst ca se therefor e is when t = k which is equiv alent to the cla ssical CUCK OO- DISJOINT hashing. W e use the unio n b ound to o bta in P f a il ( v ) < N ( v ) · p f a il ( v ) , wher e N ( v ) is the num b er of sub- graphs with v vertices, where each v e r tex of the sub-graph is hit by at least one edge, and p f a il ( v ) is the proba bilit y that a given sub-g r aph S ( V , E ) was hit by more than v edges. By definition of p f a il : (C.1) p f a il ( v ) <  m v + 1  e p v +1 hit . when t = k we get: (C.2) e p v +1 hit =  x d  xn +1 =  x d   x dx  n , and (C.3) N ( v ) ≤  n/k v /k  <  e n/k v /k  v/k =   e x  x/k  n , Since when the pag e size t equals k , an element can be placed either in all of the lo ca tions of a pag e or in none of them. (C.4)  m v + 1  <  n v + 1  <  e n v + 1  v +1 <  e n v  v +1 =  e x   e x  x  n . W e get that: (C.5) P f a il ( v ) < N ( v ) · p f a il ( v ) < N ( v )  m v + 1  e p v +1 hit < c 6 ( x ) · c n 7 ( x ) , where c 6 ( x ) = e x d − 1 and c 7 ( x ) = e ( k +1) x k x ( dk − 1 − k ) x k .  Appendix D. Proof of Lemma 12 The proof of  g a 0 ,...,a t  ≤ Q t i =0  g a i  a i is a gener alization of the pr o of giv en in [5]. F or an y p ositive in teg er t and any non-nega tive integer g : ( y 0 + ... + y t ) g = P α 0 + ... + α t = g  g α 0 ,...,α t  Q t i =0 ( y α i i ) , where the summation is taken ov er all sequences of non-neg ative int e ger indices α 0 through α t such that the sum of all α i is g . F or the sp ecial cas e where y i = a i g , a i is a non-nega tive integer and P t i =0 a i = g we get: (D.1)  g a 0 , ..., a t  t Y i =0  a i g  a i ≤ X α 0 + ... + α t = g  g α 0 , ..., α t  t Y i =0  a i g  α i = ( y 0 + ... + y t ) g = 1 So (D.2)  g a 0 , ..., a t  ≤ 1 Q t i =0  a i g  a i = t Y i =0  g a i  a i A CUCKOO HASHING V ARIANT WITH IMPRO VED MEMOR Y UTILIZA TION AND INS ER TION TIME 12 Appendix E. Proof of Lemma 13 Here we pr ov e P bad (ˆ a, β ) < O (1 ) · c 8 (ˆ a, β ) · c n 9 (ˆ a, β ) where c 8 (ˆ a, β ) = p hit (1 − p 1 − p hit ) − 1  β − x x  and c 9 (ˆ a, β ) = Q t i =0  1 ˆ a i  ˆ a i t Q t i =0  t i  ˆ a i t  β β − x  ( β − x )  β x  x p x hit (1 − p 1 − p hit ) β − x . Pr o of. Recall that ˆ a is a unit vector and v and x a re equa l to the following functions of ˆ a : (E.1) t X i =0 ˆ a i = 1 (E.2) v = t X i =0 a i · i = g t X i =0 ˆ a i · i = n t t X i =0 ˆ a i · i (E.3) x = v n = 1 t t X i =0 ˆ a i · i. P bad ( v ) < N ( v ) · p bad ( v ) , wher e (E.4) N ( v ) =  g a 0 , ..., a t  t Y i =0  t i  a i , (E.5) p bad ( v ) =  m v + 1  p v +1 hit (1 − p 1 − p hit ) m − ( v +1) , (E.6) p hit = P t i = k a i  i k  g  t k  ! d = P t i = k ˆ a i  i k   t k  ! d , (E.7) p 1 = d P t i = k ˆ a i ( t − i )  i k − 1   t k  ! P t i = k ˆ a i  i k   t k  ! d − 1 . Using lemma 12 we get: (E.8) N ( v ) =  g a 0 , ..., a t  t Y i =0  t i  a i ≤   t Y i =0  1 ˆ a i  ˆ a i t t Y i =0  t i  ˆ a i t   n , and (E.9)  m v + 1  <  m m − ( v + 1 )  m − ( v +1)  m v + 1  v +1 < O (1) ·  β β − x  ( β − x ) n − 1  β x  xn +1 . Finally we get: (E.10) P bad ( v ) < N ( v ) · p bad ( v ) = O (1) · c 8 ( x, β ) · c n 9 (ˆ a, β ) where A CUCKOO HASHING V ARIANT WITH IMPRO VED MEMOR Y UTILIZA TION AND INS ER TION TIME 13 (E.11) c 8 (ˆ a, β ) = p hit (1 − p 1 − p hit ) − 1  β − x x  , and (E.12) c 9 (ˆ a, β ) = t Y i =0  1 ˆ a i  ˆ a i t t Y i =0  t i  ˆ a i t  β β − x  ( β − x )  β x  x p x hit (1 − p 1 − p hit ) β − x .  References [1] M. Dietzfelbinger, A. Go erdt, M. M i tzenmac her, A. Montanar i , R. Pagh, and M . R i nk, “Tight thresholds for cuc koo hashing via XORSA T”. ICALP (1) 2010: 213-225 [2] E. Lehman and R. Panigra hy , “3.5-wa y cuck o o hashing for the price of 2 and a bit”. 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