Data Structures. Element Uniqueness Problem. Hash Tables. Example. Hash Tables. Dana Shapira. 19 x 1. ) h(x 4. ) h(x 2. ) h(x 3. h(x 1. x 4. x 2.

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i ]) etun; } } 2 Wht hppens when m is lge o when we e deling with el numbes?

1 Element Uniqueness Poblem Dt Stuctues Let x,..., xn < m Detemine whethe thee exist i j such tht x i =x j Sot Algoithm Bucket Sot Dn Shpi Hsh Tbles fo (i=;i<m;i++) T[i]=NULL; fo (i=;i<n;i++){ if (T[x i ]= = NULL) T[x i ]= i else{ output (i, T[x i ]) etun; } } 2 Wht hppens when m is lge o when we e deling with el numbes?? Hsh Tbles h Exmple Nottions: U univese of keys of size U, K n ctul set of keys of size n, T hsh tble of size m Use hsh-function h:u {,,m-}, h(x)=i tht computes the slot i in y T whee element x is to be stoed, fo ll x in U. h(k) is computed in O( k ) = O(). U h(x ) h(x 2 ) h(x 3 ) h(x 4 ) Set of y indices h:u {,, m-} h(x)=x mod (wht is m?) input: 7,62,9,8,53 Collision: x y but h(x) = h(y). m «U. Solutions:. Chining 2. Open ddessing x x 2 x 3 x 4 3 4

2 Collision-Resolution by Chining Anlysis of Chining Simple Unifom Hshing Any given element is eqully likely to hsh to ny slot in the hsh tble. The slot n element hshes to is independent of whee othe elements hsh. Lod fcto: α = n/m (elements stoed in the hsh tble / numbe of slots in the hsh tble) 9 Inset(T,x): Inset new element x t the hed of list T[h(x.key)]. Delete(T,x): Delete element x fom list T[h(x.key)]. Sech(T,x): Sech list T[h(x.key)]. 5 6 Anlysis of Chining Designing Good Hsh Functions Theoem: In hsh tble with chining, unde the ssumption of simple unifom hshing, both successful nd unsuccessful seches tke expected time Θ(+α) on the vege, whee α is the hsh tble lod fcto. Poof: Unsuccessful Sech: Unde the ssumption of simple unifom hshing, ny key k is eqully likely to hsh to ny slot in the hsh tble. The expected time to sech unsuccessfully fo key k is the expected time to sech to the end of list T[h(k)] which hs expected length α. expected time - θ( + α) including time fo computing h(k). Successful sech: The numbe of elements exmined duing successful sech is moe thn the numbe of elements tht ppe befoe k in T[h(k)]. n n n n i i i n + i m = n i m + = n i nm + = = = = i= n( n ) ( n ) α α α = + = + = + = + = θ ( + α ) nm 2 2m 2 2m 2 2n expected time - θ( + α) Coolly: If m = θ(n), then Inset, Delete, nd Sech tke expected constnt time. 7 Exmple: Input = els dwn unifomly t ndom fom [,) Hsh function: h(x) = Îmx Often, the input distibution is unknown. Then we cn use heuistics o univesl hshing. 8

The Division Method The Multipliction Method Hsh function: h(x) = x mod m m = 2 k h(x) = the lowest k bits of x Heuistic: m = pime numbe not too close to powe of 2 Hsh function: h(x) = Îm (cx mod ),

45 mod ) = =,.45 = 4.5... = 4 9 Efficient Implementtion of the Multipliction Method h(x) = Îm (cx mod ) Exmple h(x) = Îm (cx mod ) Let w be the size of mchine wod Assume tht key x fits into mchine

3 The Division Method The Multipliction Method Hsh function: h(x) = x mod m m = 2 k h(x) = the lowest k bits of x Heuistic: m = pime numbe not too close to powe of 2 Hsh function: h(x) = Îm (cx mod ), fo some < c < Optiml choice of c depends on input distibution. Heuistic: Knuth suggests the invese of the golden tio s vlue tht woks well: Exmple: x=23,456, m=, h(x) =, (23, mod ) = =, (76,3.45 mod ) = =,.45 = = 4 9 Efficient Implementtion of the Multipliction Method h(x) = Îm (cx mod ) Exmple h(x) = Îm (cx mod ) Let w be the size of mchine wod Assume tht key x fits into mchine wod Assume tht m = 2 p Restict ouselves to vlues of c of the fom c = s / 2 w Then cx = sx / 2 w < s < 2 w sx is numbe tht fits into two mchine wods h(x) = p most significnt bits of the lowe wod * Fctionl pt x = 23456, p = 4, m = 2 4 = 6384, w = 32, Then sx = ( ) The 4 most significnt bits of e 67; tht is, h(x) = 67 x = s = sx = h(x) = = 67 Intege pt fte multiplying by m = 2 p p bits 2

Open Addessing Line Pobing All elements e stoed diectly in the hsh tble. Lod fcto α cnnot exceed. If slot T[h(x)] is ledy occupied fo key x, we pobe ltentive loctions until we find n empty slot.

Hsh function: h(k, i) = (h'(k) + i) mod m, whee h' is n oiginl hsh function. Benefits: Esy to implement Poblem: Pimy Clusteing - Long uns of occupied slots build up s tble becomes fulle.

4 Open Addessing Line Pobing All elements e stoed diectly in the hsh tble. Lod fcto α cnnot exceed. If slot T[h(x)] is ledy occupied fo key x, we pobe ltentive loctions until we find n empty slot. Seching pobes slots stting t T[h(x)] until x is found o we e sue tht x is not in T. Insted of computing h(x), we compute h(x, i) i -the pobe numbe. Hsh function: h(k, i) = (h'(k) + i) mod m, whee h' is n oiginl hsh function. Benefits: Esy to implement Poblem: Pimy Clusteing - Long uns of occupied slots build up s tble becomes fulle. h(x) 3 4 Qudtic Pobing Double Hshing Hsh function: h(k, i) = (h'(k) + c i + c 2 i 2 ) mod m, whee h' is n oiginl hsh function. Benefits: No moe pimy clusteing Poblem: Secondy Clusteing - Two elements x nd y with h'(x) = h'(y) hve sme pobe sequence. Hsh function: h(k, i) = (h (k) + ih 2 (k)) mod m, whee h nd h 2 e two oiginl hsh functions. h 2 (k) hs to be pime w..t. m; tht is, gcd(h 2 (k), m) =. Two methods: Choose m to be powe of 2 nd guntee tht h 2 (k) is lwys odd. Choose m to be pime numbe nd guntee tht h 2 (k) < m. Benefits: No moe clusteing Dwbck: Moe complicted thn line nd qudtic pobing 5 6

Anlysis of Open Addessing Anlysis of Open Addessing Cont.

Theoem: In n open-ddess hsh tble with lod fcto α <, the expected numbe of pobes in n

Poof: Let X be the numbe of pobes in n unsuccessful sech.

nd it ccesses non-empty slot 7 8 Anlysis of Open Addessing Cont.

Theoem: Given n open-ddess hsh tble with lod fcto α <, the expected numbe of pobes in successful

5 Anlysis of Open Addessing Anlysis of Open Addessing Cont. Unifom hshing: The pobe sequence h(k, ),, h(k, m ) is eqully likely to be ny pemuttion of,, m. Theoem: In n open-ddess hsh tble with lod fcto α <, the expected numbe of pobes in n unsuccessful sech is t most / ( α), ssuming unifom hshing. Poof: Let X be the numbe of pobes in n unsuccessful sech. { } ( { } { }) { } EX [ ] = ip X= i = i P X i P X i+ = P X i i= i= i= A i = thee is n i-th pobe, nd it ccesses non-empty slot 7 8 Anlysis of Open Addessing Cont. Anlysis of Open Addessing Cont. Theoem: Given n open-ddess hsh tble with lod fcto α <, the expected numbe of pobes in successful sech is (/α) ln ( / ( α)), ssuming unifom hshing nd ssuming tht ech key in the tble is eqully likely to be seched fo. A successful sech fo n element x follows the sme pobe sequence s the insetion of element x. Conside the (i + )-st element x tht ws inseted. The expected numbe of pobes pefomed when inseting x is t most Aveging ove ll n elements, the expected numbe of pobes in successful sech is Coolly: The expected numbe of pobes pefomed duing n insetion into n open-ddess hsh tble with unifom hshing is / ( α). 9 2

Anlysis of Open Addessing Cont. Univesl Hshing A fmily H of hsh functions is univesl if fo ech pi k, l of keys, thee e t most H / m functions in H such tht h(k) = h(l).

P= ( H / m )/ H =/m This is the sme s if we chose h(k) nd h(l) unifomly t ndom fom [, m ].

6 Anlysis of Open Addessing Cont. Univesl Hshing A fmily H of hsh functions is univesl if fo ech pi k, l of keys, thee e t most H / m functions in H such tht h(k) = h(l). This mens: Fo ny two keys k nd l nd ny function h chosen unifomly t ndom, the pobbility tht h(k) = h(l) is t most /m. P= ( H / m )/ H =/m This is the sme s if we chose h(k) nd h(l) unifomly t ndom fom [, m ] Anlysis of Univesl Hshing Theoem: Fo hsh function h chosen unifomly t ndom fom univesl fmily H, the expected length of the list T[h(x)] is α if x is not in the hsh tble nd + α if x is in the hsh tble. Poof: Indicto vibles: χij = () = h( j) () h( j) hi hi Anlysis of Univesl Hshing Cont. E Y E E [ x] = χ xy = ( χ xy ) y T y T y T x y x y x y If x is not in T, then {y T : x y} = n. Hence, E[Y x ] = n / m = α. m Y x = the numbe of keys x tht hsh to the sme slot s x Y x = yt x y [ ] x χ xy EY E χ = xy y T x y 23 If x is in T, then {y T : x y} = n. Hence, E[Y x ] = (n ) / m < α. The length of list T[h(x)] is one moe, tht is, + α. 24

7 Univesl Fmily of Hsh Functions Choose pime p so tht m = p. Let x [ x,..., x ] such Fo ech [,..., ] {,..., } + = m define the hsh function s follows: H= h H = Exmple: m=p=253 =[248,223,] = xi x=25=[,2,]. m h( x) = ixi mod m i= { } 24 U = {,..., 2 } m + ( ) ( ) h x = mod 253 = 4 25 Univesl Fmily of Hsh Functions Theoem: The clss H is univesl. Poof: Let x= [ x,..., x ], y = [ y,..., y ] such tht x y, w.l.o.g Fo ll,, thee exists single such tht h( x) = h( y) h x h y = x y (mod m) x y ( ) ( ) ( ) i i i i= ( ) i( i i) x y x y (mod m) i= Fo ll z p thee exists single w such tht z w=(mod m) z = x ( ) ( ) y i xi yi x y (mod m) i= h x = h y fo m vlues ( ) ( ) The numbe of hsh functions h in H, fo which h(k ) = h(k 2 ) is t most m / m + =/m 26 Univesl Fmily of Hsh Functions Choose pime p so tht m < p. Fo ny < p nd b < p, we define function h,b (x) = ((x + b) mod p) mod m. Let H p,m be the fmily H p,m = {h,b : < p nd b < p}. Theoem: The clss H p,m is univesl. Summy Hsh tbles e the most efficient dictionies if only opetions Inset, Delete, nd Sech hve to be suppoted. If unifom hshing is used, the expected time of ech of these opetions is constnt. Univesl hshing is somewht complicted, but pefoms well even fo dvesil input distibutions. If the input distibution is known, heuistics pefom well nd e much simple thn univesl hshing. Fo collision-esolution, chining is the simplest method, but it equies moe spce thn open ddessing. Open ddessing is eithe moe complicted o suffes fom clusteing effects

Chapter 7. Kleene s Theorem. 7.1 Kleene s Theorem. The following theorem is the most important and fundamental result in the theory of FA s:

Chapter 7. Kleene s Theorem. 7.1 Kleene s Theorem. The following theorem is the most important and fundamental result in the theory of FA s: Chpte 7 Kleene s Theoem 7.1 Kleene s Theoem The following theoem is the most impotnt nd fundmentl esult in the theoy of FA s: Theoem 6 Any lnguge tht cn e defined y eithe egul expession, o finite utomt,