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1、Transform and Conquer,This group of techniques solves a problem by a transformation toa simpler/more convenient instance of the same problem(instance simplification)a different representation of the same instance(representation change)a different problem for which an algorithm is already available(p
2、roblem reduction),Instance simplification-Presorting,Solve a problems instance by transforming it intoanother simpler/easier instance of the same problemPresortingMany problems involving lists are easier when list is sorted,e.g.searching computing the median(selection problem)checking if all element
3、s are distinct(element uniqueness)Also:Topological sorting helps solving some problems for dags.Presorting is used in many geometric algorithms.,How fast can we sort?,Efficiency of algorithms involving sorting depends onefficiency of sorting.Theorem(see Sec.11.2):log2 n!n log2 n comparisons are nece
4、ssary in the worst case to sort a list of size n by any comparison-based algorithm.Note:About nlog2 n comparisons are also sufficient to sort array of size n(by mergesort).,Searching with presorting,Problem:Search for a given K in A0.n-1 Presorting-based algorithm:Stage 1 Sort the array by an effici
5、ent sorting algorithm Stage 2 Apply binary search Efficiency:(nlog n)+O(log n)=(nlog n)Good or bad?Why do we have our dictionaries,telephone directories,etc.sorted?,Element Uniqueness with presorting,Presorting-based algorithm Stage 1:sort by efficient sorting algorithm(e.g.mergesort)Stage 2:scan ar
6、ray to check pairs of adjacent elements Efficiency:(nlog n)+O(n)=(nlog n)Brute force algorithm Compare all pairs of elements Efficiency:O(n2)Another algorithm?Hashing,Instance simplification Gaussian Elimination,Given:A system of n linear equations in n unknowns with an arbitrary coefficient matrix.
7、Transform to:An equivalent system of n linear equations in n unknowns with an upper triangular coefficient matrix.Solve the latter by substitutions starting with the last equation and moving up to the first one.a11x1+a12x2+a1nxn=b1 a1,1x1+a12x2+a1nxn=b1 a21x1+a22x2+a2nxn=b2 a22x2+a2nxn=b2 an1x1+an2x
8、2+annxn=bn annxn=bn,Gaussian Elimination(cont.),The transformation is accomplished by a sequence of elementary operations on the systems coefficient matrix(which dont change the systems solution):for i 1 to n-1 do replace each of the subsequent rows(i.e.,rows i+1,n)by a difference between that row a
9、nd an appropriate multiple of the i-th row to make the new coefficient in the i-th column of that row 0,Example of Gaussian Elimination,Solve 2x1-4x2+x3=6 3x1-x2+x3=11 x1+x2-x3=-3 Gaussian elimination 2-4 1 6 2-4 1 6 3-1 1 11 row2(3/2)*row1 0 5-1/2 2 1 1-1-3 row3(1/2)*row1 0 3-3/2-6 row3(3/5)*row2 2
10、-4 1 6 0 5-1/2 2 0 0-6/5-36/5 Backward substitution x3=(-36/5)/(-6/5)=6 x2=(2+(1/2)*6)/5=1 x1=(6 6+4*1)/2=2,Pseudocode and Efficiency of Gaussian Elimination,Stage 1:Reduction to an upper-triangular matrixfor i 1 to n-1 do for j i+1 to n do for k i to n+1 do Aj,k Aj,k-Ai,k*Aj,i/Ai,i/improve!Stage 2:
11、Back substitutionsfor j n downto 1 do t 0 for k j+1 to n do t t+Aj,k*xk xj(Aj,n+1-t)/Aj,j Efficiency:(n3)+(n2)=(n3),Searching Problem,Problem:Given a(multi)set S of keys and a search key K,find an occurrence of K in S,if anySearching must be considered in the context of:file size(internal vs.externa
12、l)dynamics of data(static vs.dynamic)Dictionary operations(dynamic data):find(search)insertdelete,Taxonomy of Searching Algorithms,List searchingsequential searchbinary searchinterpolation searchTree searching binary search treebinary balanced trees:AVL trees,red-black treesmultiway balanced trees:2
13、-3 trees,2-3-4 trees,B treesHashingopen hashing(separate chaining)closed hashing(open addressing),Binary Search Tree,Arrange keys in a binary tree with the binary search tree property:,K,K,K,Example:5,3,1,10,12,7,9,Dictionary Operations on Binary Search Trees,Searching straightforwardInsertion searc
14、h for key,insert at leaf where search terminatedDeletion 3 cases:deleting key at a leafdeleting key at node with single childdeleting key at node with two children Efficiency depends of the trees height:log2 n h n-1,with height average(random files)be about 3log2 n Thus all three operations have wor
15、st case efficiency:(n)average case efficiency:(log n)Bonus:inorder traversal produces sorted list,Balanced Search Trees,Attractiveness of binary search tree is marred by the bad(linear)worst-case efficiency.Two ideas to overcome it are:to rebalance binary search tree when a new insertion makes the t
16、ree“too unbalanced”AVL trees red-black trees to allow more than one key per node of a search tree 2-3 trees 2-3-4 trees B-trees,Balanced trees:AVL trees,Definition An AVL tree is a binary search tree in which,for every node,the difference between the heights of its left and right subtrees,called the
17、 balance factor,is at most 1(with the height of an empty tree defined as-1),Tree(a)is an AVL tree;tree(b)is not an AVL tree,Rotations,If a key insertion violates the balance requirement at some node,the subtree rooted at that node is transformed via one of the four rotations.(The rotation is always
18、performed for a subtree rooted at an“unbalanced”node closest to the new leaf.),Single R-rotation,Double LR-rotation,General case:Single R-rotation,General case:Double LR-rotation,AVL tree construction-an example,Construct an AVL tree for the list 5,6,8,3,2,4,7,AVL tree construction-an example(cont.)
19、,Analysis of AVL trees,h 1.4404 log2(n+2)-1.3277 average height:1.01 log2n+0.1 for large n(found empirically)Search and insertion are O(log n)Deletion is more complicated but is also O(log n)Disadvantages:frequent rotationscomplexityA similar idea:red-black trees(height of subtrees is allowed to dif
20、fer by up to a factor of 2),Multiway Search Trees,Definition A multiway search tree is a search tree that allowsmore than one key in the same node of the tree.Definition A node of a search tree is called an n-node if it contains n-1 ordered keys(which divide the entire key range into n intervals poi
21、nted to by the nodes n links to its children):Note:Every node in a classical binary search tree is a 2-node,k1 k2 kn-1,k1,k1,k2),kn-1,2-3 Tree,Definition A 2-3 tree is a search tree that may have 2-nodes and 3-nodes height-balanced(all leaves are on the same level)A 2-3 tree is constructed by succes
22、sive insertions of keys given,with a new key always inserted into a leaf of the tree.If the leaf is a 3-node,its split into two with the middle key promoted to the parent.,2-3 tree construction an example,Construct a 2-3 tree the list 9,5,8,3,2,4,7,Analysis of 2-3 trees,log3(n+1)-1 h log2(n+1)-1Sear
23、ch,insertion,and deletion are in(log n)The idea of 2-3 tree can be generalized by allowing more keys per node 2-3-4 trees B-trees,Heaps and Heapsort,Definition A heap is a binary tree with keys at its nodes(one key per node)such that:It is essentially complete,i.e.,all its levels are full except pos
24、sibly the last level,where only some rightmost keys may be missingThe key at each node is keys at its children,Illustration of the heaps definition,a heap,not a heap,not a heap,Note:Heaps elements are ordered top down(along any path down from its root),but they are not ordered left to right,Some Imp
25、ortant Properties of a Heap,Given n,there exists a unique binary tree with n nodes that is essentially complete,with h=log2 nThe root contains the largest keyThe subtree rooted at any node of a heap is also a heapA heap can be represented as an array,Heaps Array Representation,Store heaps elements i
26、n an array(whose elements indexed,for convenience,1 to n)in top-down left-to-right orderExample:Left child of node j is at 2jRight child of node j is at 2j+1Parent of node j is at j/2 Parental nodes are represented in the first n/2 locations,9,1,5,3,4,2,Step 0:Initialize the structure with keys in t
27、he order givenStep 1:Starting with the last(rightmost)parental node,fix the heap rooted at it,if it doesnt satisfy the heap condition:keep exchanging it with its largest child until the heap condition holdsStep 2:Repeat Step 1 for the preceding parental node,Heap Construction(bottom-up),Example of H
28、eap Construction,Construct a heap for the list 2,9,7,6,5,8,Pseudopodia of bottom-up heap construction,Stage 1:Construct a heap for a given list of n keysStage 2:Repeat operation of root removal n-1 times:Exchange keys in the root and in the last(rightmost)leafDecrease heap size by 1If necessary,swap
29、 new root with larger child until the heap condition holds,Heapsort,Sort the list 2,9,7,6,5,8 by heapsortStage 1(heap construction)Stage 2(root/max removal)1 9 7 6 5 8 9 6 8 2 5 7 2 9 8 6 5 7 7 6 8 2 5|9 2 9 8 6 5 7 8 6 7 2 5|9 9 2 8 6 5 7 5 6 7 2|8 9 9 6 8 2 5 7 7 6 5 2|8 9 2 6 5|7 8 9 6 2 5|7 8 9
30、5 2|6 7 8 9 5 2|6 7 8 9 2|5 6 7 8 9,Example of Sorting by Heapsort,Stage 1:Build heap for a given list of n keysworst-case C(n)=Stage 2:Repeat operation of root removal n-1 times(fix heap)worst-case C(n)=Both worst-case and average-case efficiency:(nlogn)In-place:yesStability:no(e.g.,1 1),2(h-i)2i=2
31、(n log2(n+1)(n),i=0,h-1,#nodes at level i,i=1,n-1,2log2 i(nlogn),Analysis of Heapsort,A priority queue is the ADT of a set of elements with numerical priorities with the following operations:find element with highest prioritydelete element with highest priorityinsert element with assigned priority(s
32、ee below)Heap is a very efficient way for implementing priority queuesTwo ways to handle priority queue in which highest priority=smallest number,Priority Queue,Insertion of a New Element into a Heap,Insert the new element at last position in heap.Compare it with its parent and,if it violates heap c
33、ondition,exchange themContinue comparing the new element with nodes up the tree until the heap condition is satisfiedExample:Insert key 10Efficiency:O(log n),Horners Rule For Polynomial Evaluation,Given a polynomial of degree np(x)=anxn+an-1xn-1+a1x+a0and a specific value of x,find the value of p at
34、 that point.Two brute-force algorithms:p 0 p a0;power 1for i n downto 0 do for i 1 to n do power 1 power power*x for j 1 to i do p p+ai*power power power*x return p p p+ai*powerreturn p,Horners Rule,Example:p(x)=2x4-x3+3x2+x-5=x(2x3-x2+3x+1)-5=x(x(2x2-x+3)+1)-5=x(x(x(2x-1)+3)+1)-5 Substitution into
35、the last formula leads to a faster algorithm Same sequence of computations are obtained by simply arranging the coefficient in a table and proceeding as follows:coefficients2-1 3 1-5 x=3,Horners Rule pseudocode,Efficiency of Horners Rule:#multiplications=#additions=n Synthetic division of of p(x)by(
36、x-x0)Example:Let p(x)=2x4-x3+3x2+x-5.Find p(x):(x-3),Computing an(revisited),Left-to-right binary exponentiation Initialize product accumulator by 1.Scan ns binary expansion from left to right and do the following:If the current binary digit is 0,square the accumulator(S);if the binary digit is 1,sq
37、uare the accumulator and multiply it by a(SM).Example:Compute a13.Here,n=13=11012 binary rep.of 13:1 1 0 1 SM SM S SM accumulator:1 12*a=a a2*a=a3(a3)2=a6(a6)2*a=a13(computed left-to-right)Efficiency:b M(n)2b where b=log2 n+1,Computing an(cont.),Right-to-left binary exponentiation Scan ns binary exp
38、ansion from right to left and compute an as the product of terms a2 i corresponding to 1s in this expansion.Example Compute a13 by the right-to-left binary exponentiation.Here,n=13=11012.1 1 0 1 a8 a4 a2 a:a2 i terms a8*a4*a:product(computed right-to-left)Efficiency:same as that of left-to-right bin
39、ary exponentiation,Problem Reduction,This variation of transform-and-conquer solves a problem by a transforming it into different problem for which an algorithm is already available.To be of practical value,the combined time of the transformation and solving the other problem should be smaller than
40、solving the problem as given by another method.,Examples of Solving Problems by Reduction,computing lcm(m,n)via computing gcd(m,n)counting number of paths of length n in a graph by raising the graphs adjacency matrix to the n-th powertransforming a maximization problem to a minimization problem and vice versa(also,min-heap construction)linear programmingreduction to graph problems(e.g.,solving puzzles via state-space graphs),
