Probabilistic analysis of algorithms: What s it good for?
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1 Probabilistic analysis of algorithms: What s it good for? Conrado Martínez Univ. Politècnica de Catalunya, Spain February 2008
2 The goal Given some algorithm taking inputs from some set Á, we would like to analyze the performance of the algorithm as a function of the input size (and possibly other parameters).
3 Why should we analyze algorithms? To predict the resources (time, space.... ) that the algorithm will consume To compare algorithm with competing alternatives To improve the algorithm, by spotting the performance bottlenecks To explain observed behavior
4 Why should we analyze algorithms? To predict the resources (time, space.... ) that the algorithm will consume To compare algorithm with competing alternatives To improve the algorithm, by spotting the performance bottlenecks To explain observed behavior
5 Why should we analyze algorithms? To predict the resources (time, space.... ) that the algorithm will consume To compare algorithm with competing alternatives To improve the algorithm, by spotting the performance bottlenecks To explain observed behavior
6 Why should we analyze algorithms? To predict the resources (time, space.... ) that the algorithm will consume To compare algorithm with competing alternatives To improve the algorithm, by spotting the performance bottlenecks To explain observed behavior
7 Concepts & definitions The performance Á Æ depends on each particular instance of the input We have to introduce some notion of size: Á Æ; we may safely assume that each Á Ò Ü ¾ Á Ü Ò is finite Worst case: [worst] Òµ ÑÜ Üµ Ü ¾ Á Ò
8 To analyze typical behavior or the performance of randomized algorithms, we have to assume some probabilistic distribution on the input and/or the algorithm s choices; hence, we consider the performance as a family of random variables Ò Ò¼ ; Ò Á Ò Æ Average case: [avg] Òµ Ò ¼ È Ò
9 When we assume uniformly distributed inputs ÈÜ ½ Á Ò for all Ü ¾ Á Ò our problem is one of counting, e.g., È Ü¾ÁÒ Ò Üµ Á Ò
10 One of the most important tools in the analysis of algorithms are generating functions: Þ Ùµ Ò¼ ¼ È Ò Þ Ò Ù For the uniform distribution Þ Ò Ù Þ Ùµ Þ Ò Ù È Ò¼ ȼ È ÒÞ Ò Ù Þ Ò Ò¼ ÒÞ Ò ÞÒ Ù Þ Ùµ Þ Ò Þ ½µ with Ò Ü ¾ Á Ü Ò Üµ and Ò Á Ò
11 The equations before can be expressed symbolically Þ Ùµ ܾÁ Þ Ü Ù Üµ The ratio of the Ò th coefficients of Þ Ùµ and Þ ½µ is the probability generating function of Ò Ô Ò Ùµ ¼ È Ò Ù ÞÒ Þ Ùµ Þ Ò Þ ½µ
12 Taking derivatives w.r.t. Ù and setting Ù ½ we get the expected value, second factorial moment,... Öµ Þµ Ö Þ Ùµ Ù Ö Ò¼ Ù½ Ò Ö Þ Ò For example, Î Ò Ò ¾ Ò Ò ¾ Þ Ò ¾µ Þµ Þ Ò Þµ Þ Ò Þµµ ¾
13 The symbolic method The symbolic method translates combinatorial constructions to functional equations over generating functions. Example: Consider the counting generating function of a combinatorial class : If then Þµ «¾ Þ «Þµ Þµ Þµ Ò¼ µ¾ Ò Þ Ò Þ «¾ Þ «¼ ¾ ½ Þ ¼ ¾ Þ ½
14 A dictionary of (labelled) combinatorial constructions and G.F. s ËÕ µ ½ Þ ½ ½ ËØ µ ÜÔ µ ÝÐ µ ÐÓ ½ ½
15 Complex analysis Another important set of techniques comes from complex variable analysis. Under suitable technical conditions, if Þµ is analytic in a disk Þ ¾ Þ ½ and has a single dominant singularity at Þ ½ then Þµ Þµ µ Þ Ò Þµ Þ Ò Þµ This is one of the useful transfer lemmas of Flajolet and Odlyzko (1990). Many other similar results are extremely useful when computing asymptotic estimates for the Ò th coefficient of a generating function.
16 For example, if Þµ Þµ ½ Þ «À Þµ as Þ, the dominant singularity of Þµ, for some analytic Þµ and À Þµ and «¾ ½ ¾ then Þ Ò Þµ µ Ò Ò«½ «µ ½ Ç Ò ½ µ
17 Complex analysis In recent years, complex analysis techniques and perturbation theory have been used to prove powerful results such as Hwang s quasi power theorem, which allows one to prove the convergence in law to a Gaussian distribution of many combinatorial parameters in strings, permutations, trees, etc., as well as local limits and the speed of convergence.
18 A trivial example: Counting binary trees A binary tree is either an empty tree (leaf) or a root together with two binary (sub)trees: Hence the counting GF of binary trees is Þµ ½ Þ ¾ Þµ
19 Solving the equation before for Þµ and since ¼µ ¼ ½, Þµ ½ Ô ½ Þ ¾Þ ½ Þ ¼ Þ ¼ Extracting the Ò th coefficient of Þµ we find ¾Ò Ò Þ Ò Þµ Ò ½ Ò ¾ Ò Ô
20 2 nd example: The average cost of building BSTs A binary search tree Ì for a set of elements contains some Ý ¾ at its root; its subtrees Ä and Ê are binary search trees recursively constructed for the sets Ü ¾ Ü Ý and Þ ¾ Ý Þ. Binary search trees (BSTs) are useful for fast lookup, and support both dynamic insertions and deletions.
21 To insert some new item Û into a BST, we compare Û to the element Ý at the root of Ì. If Û Ý then we insert Û recursively in the left subtree of Ì. If Ý Û we insert recursively in the right subtree. If at any point we have to insert the element into an empty tree, we simply make Û the root of the new tree
22 How much does it cost to build a BST of size Ò? Let Ü Ì µ denote the depth (= edges from the root) of element Ü in Ì. It is the number of elements with which Ü was compared when we inserted it at some previous step. Hence, the cost Á Ì µ to build Ì is Á Ì µ Ü¾Ì Ü Ì µ Á Ì µ is also called the internal path length of Ì.
23 Let Ò Ì. If Ì is a linked list (at most one non empty subtree per node) then Á Ì µ Ò Ò ½µ¾. If Ì is perfectly balanced then Á Ì µ Ò ÐÓ Òµ.
24 In a random BST any element has the same probability of being its root; hence the probability that Ä is ½Ò for all, ¼ Ò. ÈÌ ½ ÈÄ ÈÊ Ì if Ì is empty, otherwise.
25 Á Ì µ ¼ Á ĵ Á ʵ Ì ½ if Ì is empty, otherwise.
26 Á Ò Þ Ò Á Þµ Á Þµ Ì ¾ ÈÌ Á Ì µþ Ì Äʵ¾ ÈÄ È Ê Á ĵ Á ʵ Ä ÊµÞÄ Ê ½ Ä Ê ½
27 Á Þ Äʵ¾ Äʵ¾ Äʵ¾ Äʵ¾ Äʵ¾ ÈÄ È Ê Á ĵ Á ʵ Ä ÊµÞ Ä Ê ÈÄ È Ê Á ÄµÞ Ä Ê ÈÄ È Ê Á ÊµÞ Ä Ê ÈÄ È Ê ÄÞ Ä Ê ÈÄ È Ê ÊÞ Ä Ê
28 Á Þ ¾ ¾ ¾ ¾ Ì ½Ì ¾µ¾ ÈÌ ½ Á Ì ½ µþ ̽ È Ì ¾ Þ Ì¾ ¼ Ì ½Ì ¾µ¾ ÈÌ ½ Ì ½ Þ Ì½ È Ì ¾ Þ Ì¾ ¼ Ì ½¾ Ì ½¾ ÈÌ ½ Á Ì ½ µþ ̽ ½ ¼ Ì ¾¾ ÈÌ ½ Ì ½ Þ Ì½ ½ ¼ Ì ¾¾ ÈÌ ¾ Þ Ì¾ ½ ÈÌ ¾ Þ Ì¾ ½
29 ÈÌ Þ Ì Ì ¾ Ò¼ Þ Ò ½ ½ Þ Ì ¾ ÈÌ Ì Þ Ì Ò¼ ÒÞ Ò Þ ½ Þ ½ Þ Þ ½ Þµ ¾
30 Á Þ ¾ Á Þµ ½ Þ ÐÒ Á Þµ ¾ Á ¼µ ¼ ½ ½ Þ ¾Þ ½ Þµ ¾Þ ½ Þµ ¾ ½ Þµ ¾
31 Á Ò Þ Ò Á Þµ ¾ Ò ½µÀ Ò ¾Ò ÐÒ Ò ¾Ò Ç ÐÓ Òµ À Ò ½Ò ½ ÐÒ Ò Ç Ò ½ µ
32 3 rd example: Insertion in à d trees
33 3 rd example: Insertion in à d trees 1 1
34 3 rd example: Insertion in à d trees
35 3 rd example: Insertion in à d trees
36 3 rd example: Insertion in à d trees
37 3 rd example: Insertion in à d trees
38 Insertion in relaxed à d trees ÖØ Ò ÖØ ÖØ Ø ÓÒ Ø ÐÑ ² Ü µ ß ÒØ Ò Þ Ø µ ÒØ Ù ÖÒÓÑ ¼ Ò µ Ù Ò µ ÖØÙÖÒ Ò ÖØØÖÓÓØ Ø Ü µ Ð ß»» Ø ÒÒÓØ ÑÔØÝ ÒØ Ø ¹ Ö Ü Ø ¹ Ý µ Ø ¹ ÐØ Ò ÖØ Ø ¹ ÐØ Ü µ Ð Ø ¹ ÖØ Ò ÖØ Ø ¹ ÖØ Ü µ ÖØÙÖÒ Ø Ð Ð
39 Deletion in relaxed à d trees ÖØ ÐØ ÖØ Ø ÓÒ Ø ÐÑ ² Ü µ ß Ø ÆÍÄÄ µ ÖØÙÖÒ ÆÍÄÄ Ø ¹ Ý Üµ ÖØÙÖÒ ÓÒ Ø ¹ ÐØ Ø ¹ ÖØ µ Ð ÒØ Ø ¹ Ö Ü ¹ Ý Ø ¹ Ý µ Ø ¹ ÐØ ÐØ Ø ¹ ÐØ Ü µ Ð Ø ¹ ÖØ ÐØ Ø ¹ ÖØ Ü µ ÖØÙÖÒ Ø
40 Split: Case #1 t r
41 Split: Case #1 t r
42 Split: Case #1 t r
43 Split: Case #2 t r
44 Split: Case #2 t r
45 Split: Case #2 t r
46 Split: Case #2 t r
47 Analysis of split/join Ò avg. number of visited nodes in a split Ñ Ò avg. number of visited nodes in a join Ò ½ ¾ ÒÃ Ã ½ Ã ¼Ò ¼Ò ½ Ò ½ ¾ Ã ½µ ÒÃ Ò Ñ ¼Ò where Ò is probability of joining two trees with total size.
48 Analysis of split/join Ò avg. number of visited nodes in a split Ñ Ò avg. number of visited nodes in a join Ò ½ ¾ ÒÃ Ã ½ Ã ¼Ò ¼Ò ½ Ò ½ ¾ Ã ½µ ÒÃ Ò Ñ ¼Ò where Ò is probability of joining two trees with total size.
49 Analysis of split/join Ò avg. number of visited nodes in a split Ñ Ò avg. number of visited nodes in a join Ò ½ ¾ ÒÃ Ã ½ Ã ¼Ò ¼Ò ½ Ò ½ ¾ Ã ½µ ÒÃ Ò Ñ ¼Ò where Ò is probability of joining two trees with total size.
50 The recurrence for Ò is Ò ½ ¾ ÒÃ with ¼ ¼. ¼Ò ¾ Ã ½µ ÒÃ ½ Ò ½ ¾ Ã ½µ ÒÃ ¼Ò Ò Ò ½ Ñ ¼Ò The recurrence for Ñ Ò has exactly the same shape with the rôles of Ò and Ñ Ò interchanged; it easily follows that Ò Ñ Ò.
51 Define Ë Þµ Ò¼ Ò Þ Ò The recurrence for Ò translates to Þ ¾ Ë Þ ¾ ¾ ½ ¾Þ ½ Þ Ë Þ Ã ¾ ¾ Ã Ë Þµ Þ ½ Þµ ¾ ¾ ½ Þµ with initial conditions Ë ¼µ ¼ and Ë ¼ ¼µ ½.
52 The homogeneous second order linear ODE is of hypergeometric type. An easy particular solution of the ODE is ½ Ã ½ ¾ Ã ½ ½ Þ
53 Theorem The generating function Ë Þµ of the expected cost of split is, for any à ¾, ½ ½ Ë Þµ ½ ¾ ½ ½ à where ««Ãµ ½ ¾ ½ Þµ «¾ ½ ½ «¾ «¾ Õ ½ ½ ½. Ã Þ ½ Þ
54 Theorem The expected cost Ò of splitting a relaxed à d tree of size Ò is Ò Ãµ Ò Ãµ Ó Òµ with ½ ¾ ½ ½ ½ à «½ ½ ¾ ¾«½µ ««µ ¼ ½ ½ à ½ ½
55 K Plot of õ
56 K Plot of õ
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