Binary Heaps A binary heap is dened to b e a binary tree with a key in each no de such that All leaves are on at most twoadjacent levels All leaves on the lowest level occur to the left and all levels except the lowest one are completely lled The key in ro ot is all its children and the left and right subtrees are again binary heaps Conditions and sp ecify shap e of the tree and con dition the lab eling of the tree 1/1 7/4 2/2 12/7 10/30 12/25 The ancestor relation in a heap denes a partial or der on its elements which means it is reexive anti symmetric and transitive Reexive x is an ancestorof itself Antisymmetric if x is an ancestor of y and y is an ancestorof xthenx y Transitive if x is an ancestor of y and y is an ancestorofz x is an ancestorofz Partial orders can be used to mo del heirarchies with incomplete information or equalvalued elements One of myfavorite games with myparents is eshing out the partial order of big oldtimemovie stars The partial order dened by the heap structure is weaker than that of the total order which explains Why it is easier to build Why it is less useful than sorting but still very important Constructing Heaps Heaps can be constructed incrementally by inserting new elements into the leftmost op en sp ot in the array If the new element is greater than its parent swap their p ositions and recur Since at each step wereplace the ro ot of a subtree by alarger one wepreserve the heap order Since all but the last level is always lled the height h of an n element heap is b ounded b ecause h X i h n i so h blg nc Doing n such insertions takes n log n since the last n insertions require O log N time each Heapify The b ottom up insertion algorithm gives a good way to build a heap but Rob ert Floyd found a b etter way using a merge pro cedure called heapify Given two heaps and a fresh element they can be merged into one bymaking the new one the ro ot and trickling down BuildheapA n jAj For i bnc todo HeapifyAi HeapifyAi left i right i if lef t nandAlef t Ai then max left else max i if right n and Aright Amax then max right if max ithen swapAiAmax HeapifyAmax Rough Analysis of Heapify Heapify on a subtree containing n no des takes T n T n O The comes from merging heaps whose levels dif fer by one The last row could be exactly half lled Besides the asymptotic answer wont change so long the fraction is less than one Solve the recurrence using the Master Theorem Let a b and f n log Note that n since log Thus Case of the Master theorem applies The Master Theorem Let a andbbeconstants let f n be a function and let T nbedened on the nonnegative integers by the recurrence T naT nbf n where we interpret nb to mean either bnbc or dnbe Then T n can b e b ounded asymptotically as follows log a b If f nO n forsomeconstant then T n log a b n log a log a b b If f nn then T nn lg n log a b If f n n for some constant and if af nb cf nforsomeconstant c and all suciently large n then T nf n Exact Analysis of Heapify In fact Heapify performs b etter than O n log n be cause most of the heaps wemerge are extremely small A Z T M Y RJ CDBX LPFH In a full binary tree on n no des there are n no des which are leaves ie height nnodes which are height nnodes which are height h In general there are at most dn e no des of height hso the cost of building a heap is blg nc blg nc X X h h h dn eO hO n h h Since this sum is not quite a geometric series we cant apply the usual identitytogetthe sum But it should be clear that the series converges Pro of of Convergence Series convergence is the free lunch of algorithm analysis The identify for the sum of a geometric series is X k x x k If we takethe derivative of b oth sides X k kx x k Multiplying both sides of the equation by x gives the identitywe need X x k kx x k Substituting x gives a sum of so Buildheap uses at most n comparisons and thus linear time The Lessons of Heapsort I Are we doing a careful analysis Might our algorithm be faster than it seems Typically in our analysis wewillsaythat since weare doing at most x op erations of at most y timeeach the total timeis O xy However if we overestimate to o much our b ound may not b e as tight as it should b e Heapsort Heapify can be used to construct a heap using the observation that an isolated element formsa heap of size HeapsortA BuildheapA for i n todo swapAAi n n HeapifyA If we construct our heap from bottom to top using Heapify we do not have to do anything with the last nelements With the implicit tree dened byarraypositions ie the ith p osition is the parent of the ith and i st p ositions the leaves start out as heaps Exchanging the maximum element with the last ele ment and calling heapify rep eatedly gives an O n lg n sorting algorithmnamed Heapsort Heapsort Animations The Lessons of Heapsort II Always ask yourself Can weuse a dierent data struc ture Selection sort scans throught the entire array rep eat edly nding the smallest remaining element For i to n A Find the smallest of the rst n i items B Pull it out of the arrayand put it rst Using arrays orunsorted linked lists as the data struc ture op eration A takes O n time and op eration B takes O Using heaps both of these op erations can be done within O lg ntime balancing the work and achieving abetter tradeo Priority Queues A priority queue is a data structure on sets of keys supp orting the following op erations InsertS x insert x into set S MaximumS return the largest key in S ExtractMaxS return and remove the largest key in S These op erations can b e easily supp orted using a heap Insert use the trickle up insertion in O log n Maximum read the rst element in the arrayin O ExtractMax delete rst element replace it with the last decrement the element counter then heapify in O log n Applications of Priority Queues Heaps as stacks or queues In a stack push inserts a new item and pop re moves the most recently pushed item In a queue enqueue inserts a new item and de queue removes the least recently enqueued item Both stacks and queues can be simulated by using a heap when weaddanew time eld to each item and order the heap according it this timeeld To simulate the stack increment the time with each insertion and put the maximum on top of the heap To simulate the queue decrement the time with each insertion and put the maximum on top of the heap or increment times and keep the minimum on top This simulation is not as ecient as a normalstackqueue implementation but it is a cute demonstration of the exibilityofapriorityqueue Discrete Event Simulations In simulations of airp orts parking lots and jaialai priority queues can b e used to maintain who go es next The stack and queue orders are just sp ecial cases of orderings In real life certain p eople cut in line Sweepline Algorithms in Computational Geometry In the priority queue wewill store the p oints wehave not yet encountered ordered by x co ordinate and push the line forward one stop at a time.