quality/quantity peak time/ratio

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1 Semi-Heap ad Its Applicatios i Touramet Rakig Jie Wu Departmet of omputer Sciece ad Egieerig Florida Atlatic Uiversity oca Rato, FL 3343 jie@cse.fau.edu September, 00

2 . Itroductio ad Motivatio. relimiaries 3. Semi-Heap 4. Geeralized Sortig Usig Semi-Heap 5. arallel Geeralized Sortig Usig Semi-Heap. Other Results 7. oclusios

3 Itroductio Two dieret worlds Hardyism Utility as a goal is iferior to elegace ad profudity. Maoism Scietic research should serve proletaria polities,..., ad be itergrated with productive labor. 3

4 Golde-ratio-based Search: Golde ratio: = p 5, = 0:::: quality/quatity peak time/ratio Questios: Why golde-ratio-based search? Golde-ratio-based search or biary-tree-based search? 4

5 Touramet: players where every possible pair of players plays oe game to decide the wier (ad the loser) betwee them. Graph represetatio: A directed graph with a complete uderlyig graph. 3 5 Hamiltoia path (also called geeralized sorted sequece): 3 5 Lower boud: ( log ) Sortig algorithms: bubble sort, biary isertio sort, merge sort We exted a heapsort algorithm usig a semi-heap ad the geeralize it to a cost-optimal parallel algorithm uder the EREW RAM model with () ru time usig (log ) processors. 5

6 Touramet Rakig roblem: geeralized sorted sequece (( log )) sorted sequece of kigs (Wu 000, cojectured to be ( )) local media order (O( )) media order (N-complete) kig: other players are beate by the kig directly or idirectly (via a third player). media order: rakig of players with a miimum umber of total upsets. geeralized sorted sequece sorted sequece of kigs local media order media order

7 relimiaries Touramet Existece of a Hamiltoia path i ay touramet: repositio: osider a set N (jnj = ) with ay two elemets i ad j, either i j or j i. Elemets i N ca be arraged i a liear order 0 0 ::: 0, 0 Assume that the prepositio holds for = k: 0 0 ::: 0 k Whe = k +, we isert the (k + )th elemet 0 k+ i frot of 0 i, where i is the smallest idex such that 0 k+ 0 i : 0 0 ::: 0 k+ 0 ::: 0 i k If such a idex i does ot exist, 0 k+ is placed as the last elemet: 0 0 ::: 0 k 0 k+ 7

8 Heap Heap is a array A that ca be viewed as a complete biary tree. The left child of A[i] is A[l(i)] = A[i] ad the right child of A[i] is A[r(i)] = A[i + ]. Heap property: For every ode i other tha the root: A[ aret(i)] A[i] i i i+ heapsize A[l[i)] A[i] A[r(i)]... A[i]... A[l(i)] A[r(i)]... *... * (a) (b)

9 Faculty Recrutig rocess: Yes/No Offer selected cadidate cadidates cadidate pool A k-roud selectio process ost fuctio: Type Radom Sorted Heap ostruct () ( log ) () Select () () () Maitai () () (log ) Overall cost: Type Radom Sorted Heap k = () () ( log ) () k =() ( ) ( log ) ( log ) k =(= log ) ( ) ( log ) () 9

10 3 Semi-Heap Deitio : = maxf ; ; 3 g if both = maxf ; ; 3 g ad 3 = maxf ; ; 3 g are false. Four possible coguratios of a triagle i a semi-heap. A[i] A[i] A[l(i)] A[r(i)] A[l(i)] A[r(i)] (a) (b) A[i] A[i] A[l(i)] A[r(i)] A[l(i)] A[r(i)] (c) (d) Deitio : A semi-heap for a give itrasitive total order is a complete biary tree. For every ode 0 i the tree, 0 = maxf 0 ;L( 0 );R( 0 )g. 0

11 ostruct a semi-heap from a radom array: SEMI-HEAIFY(A; i) costructs a semi-heap rooted at A[i], provided that biary trees rooted at A[l(i)] ad A[r(i)] are semi-heaps. (Its cost is (log ), where = heapsize.) UILD-SEMI-HEA(A) uses the procedure SEMI-HEAIFY i a bottomup maer to covert a arbitrary array A ito a semi-heap. (Its cost is ()) SEMI-HEAIFY(A; i) if A[i] = maxfa[i];a[l(i)];a[r(i)]g the d wier such that A[wier], maxfa[i];a[l(i)];a[r(i)]g 3 exchage A[i]! A[wier] 4 SEMI-HEAIFY(A; wier) UILD-SEMI-HEA(A) for i, b heapsize c dowto do SEMI-HEAIFY(A; i)

12 The descriptio of the SEMI-HEAIFY algorithm: A[i] semi-heap A[l(i)] A[r(i)] A[l(l(i)] semi-heap A[r(l(i))] semi-heap

13 Theorem : UILD-SEMI-HEA costructs a semi-heap for ay give complete biary tree. A example of usig UILD-SEMI-HEA: A: *... * A: *... * (a) (b) 3

14 4 Geeralized Sortig Usig Semi-Heap Why the traditioal heapsort caot be used? With the trasitive property, root A[] \beats" all the other \players". Whe the root is discarded, it is replaced by the last elemet A[] i the heap. The the heap is recostructed by pushig A[] dow i the heap if ecessary so that the ew root is the maximum elemet amog the remaiig oes. I a semi-heap, the followig situatio may occur: A[] \beats" all A[], A[], ad A[3]. A[] A[]... A[3]... A[] 4

15 Geeralized sortig usig semi-heap Geeralized sortig is doe through SEMI-HEA-SORT by repeatly pritig ad removig the root of the biary tree (which is iitially a semi-heap). The root is replaced by either its left child or right child through RE- LAE. The selected child is replaced by oe of its childre. The process cotiues util a leaf ode is reached ad the etry for the leaf ode is replaced by. 5

16 RELAE(A; i) repeatly replaces a ode (startig from the root) by either its leftchild or rightchild util the curret ode is a leave ode. (Its cost is bouded the height of the origial semi-heap, (log )). SEMI-HEA-SORT repeatly prits ad removes the root of the biary tree (which is iitially a semi-heap). (Its cost is ( log ).) RELAE(A; i) if (A[l(i)] = ) ^ (A[r(i)] = ) the A[i], 3 else if (A[i] A[l(i)]) ^ (A[l(i)] A[r(i)]) 4 the A[i], A[l(i)] 5 RELAE(A; l[i]) else A[i], A[r(i)] 7 RELAE(A; r[i]) SEMI-HEA-SORT(A) UILD-SEMI-HEA(A) while (A[l()] = ) _ (A[r()] = ) 3 do prit(a[]) 4 RELAE(A, ) 5 prit(a[])

17 Theorem : For ay give semi-heap, SEMI-HEA-SORT geerates a geeralized sorted sequece. Touramet represetatio: A touramet is represeted by a matrix M. M[i; j] = if i beats j (i.e., i j ). M[i; j] =0 if i is beate by j (i.e., j i ). M[i; i] =, represets a impossible situatio. M = 0 0 0, , , 0 0 0, , , , A 7

18 A step-by-step applicatio of RELAE(A; i): * *... * * * *... * (a) (b) * * * *... * * 4 5 * * * *... * (c) (d) * * 5 * * * *... * * * * * * * *... * (e) (f)

19 5 arallel Geeralized Sortig Usig Semi-Heap A cost-optimal parallel algorithm A sortig algorithm is cost-optimal if the product of ru time ad the umber of processors is ( log ). RELAE(A; ) is pipelied level to level ad this statemet is called at every other step (sice each ode is shared by two processors at adjacet level, a idle step is iserted betwee two calls). The ru time of SEMI-HEA-SORT is reduced to () with (log ) processors. This parallel algorithm rus o the REW RAM model, but ca be easily modied to the EREW RAM model without additioal cost. 9

20 From REW RAM to EREW RAM: resolve memory access coict The etwork model: a liear array of processors 0,,,... h, where h = dlog e. 0

21 Active ad passive steps: active step passive step Step k i- i i+ Step k i- i i+ At a eve step, processors 0,, 4,... take the active step ad processors, 3, 5,... take the passive step. The role of active ad passive amog these processors exchages i the ext step.

22 Theorem 3: The proposed parallel implemetatio is cost-optimal with a ru time of () usig (log ) processors. 0 at a active step (starts from step 0):. rits root A[].. If both child odes are, A[] is replaced by ad the 0 seds a termiatio sigal to ad stops. If at least oe child ode is ot, replaces A[] by oe of twochild odes, A[] ad A[3], followig the rule i RELAE. If A[] is selected, 0 seds id = toprocessor ; otherwise, id = 3 is set. I the ext step (a passive step), 0 receives (id; replacemet) from, ad the, performs the update A[id] :=replacemet.

23 i, i = 0, at a passive step: If i receives (id; replacemet) from i+, it performs the update A[id] := replacemet. If i receives sigal id = j from i,, it performs the followig activities i ext active step:. If both childre are, A[j] is replaced by ; otherwise, A[j] is replaced by either A[j] or A[j + ] based the replacemet rule.. Sed (j; A[j]) to i,. 3. If either A[j] or A[j + ] is selected to replace A[j], the correspodig id (j or j +)is set to i+, provided i is ot the last processor (i.e., i = h,); otherwise, the selected elemet is replaced by. If i receives the termiatio sigal, it forwards the termiatio sigal to the ext processor i+ (if it exists) i the ext active step, ad the, i stops. 3

24 A step-by-step illustratio: (3, ) (a) (b) (3, ) 3 3 (7, *) * * 5 * 5 * (c) (d) (, ) 4 3 (, *) * * * * 5 * * 5 * * (e) (f) 4

25 Other Results Sorted Sequece of Kigs: Quicksort i(u): a set of players that beat u. out(u): a set of players that are beate by u. i(u) u out(u) 5

26 arallel Merge: A EREW RAM model with ruig time O(log ) usig O(= log ) processors. p split p split q q first= (a) last=m first= (b) last=m p split p split q first= (c) last=m q first= cut (d) cut+ last=m

27 7 oclusios A data structure called semi-heap. A optimal solutio to the geeralized sortig problem. A cost-optimal EREW RAM algorithm with () i ru time usig (log ) processors. A implemetatio of the proposed parallel algorithm uder the etwork model usig a liear array of processors. 7