# Concept: Types of algorithms

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2 1002 Cocept: Types of algorithms ad algorithm aalyses, by Kut Reiert, 18. Oktober 2010, 21:22 A skier must decide every day she goes skiig, whether to ret or to buy skis, uless or util she decides to buy them. The skier does ot kow how may days she ca ski, because the whether is upredictable. Call the umber of days she will ski T. The cost to ret skis is 1 uit, while the cost of buyig skis is B. What is the optimal offlie algorithm miimizig the worst case cost? Ad what would be the optimal strategy i the olie case? Exact vs approximate vs. heuristic vs. operatioal Usually algorithms have a optimizatio goal i mid, e.g. compute the shortest path or the aligmet or miimal edit distace. Exact algorithms aim at computig the optimal solutio give such a goal. Ofte this is quite expesive i terms of ru time or memory ad hece ot possible for large iput. I such cases oe tries other strategies. Approximatio algorithms aim at computig a solutio which is for example oly a certai, guarateed factor worse tha the optimal solutio, that meas a algorithm yields a c approximatio, if it ca guaratee that its solutio is ever worse tha a factor c compared to the optimal solutio. Alteratively, heuristic algorithms try to reach the optimal solutio without givig a guaratee that they always do. Ofte it is easy to costruct a couter example. A good heuristics is almost always ear or at the optimal value. Fially there are algorithms which do ot aim at optimizig a objective fuctio. I call them operatioal sice they chai a series of computatioal operatios guided by expert kowledge but ot i cojuctio with a specific objective fuctio (e.g. ClustalW). Example: Approximatio algorithm As a example thik of the Travelig Salema Problem with triagle iequality for cities. This is a NP-hard problem (o polyomial-time algorithm is kow). The followig greedy, determiistic algorithm yields a 2 approximatio for the TSP with triagle iequality i time O( 2 ). 1. Compute a miimum spaig tree T for the complete graph implied by the cities. 2. Duplicate all edges of T yieldig a Euleria graph T ad the fid a Euleria path i T. 3. Covert the Euleria cycle ito a Hamiltoia cycle by takig shortcuts. Ca you ow argue why this is a 2 approximatio? Categorizatio accordig to mai cocept Aother way which you have ofte heard util ow is to use the mai algorithmic paradigm to categorize a algorithm, such as: Simple recursive algorithms Backtrackig algorithms Divide-ad-coquer algorithms Dyamic programmig algorithms Greedy algorithms Brach-ad-boud algorithms

3 Cocept: Types of algorithms ad algorithm aalyses, by Kut Reiert, 18. Oktober 2010, 21: Brute force algorithms ad others... Simple recursive algorithms A simple recursive algorithm Solves the base cases directly Recurs with a simpler subproblem Does some extra work to covert the solutio to the simpler subproblem ito a solutio to the give problem Examples are: To cout the umber of elemets i a list: If the list is empty, retur zero; otherwise, Step past the first elemet, ad cout the remaiig elemets i the list Add oe to the result To test if a value occurs i a list: If the list is empty, retur false; otherwise, If the first thig i the list is the give value, retur true; otherwise Step past the first elemet, ad test whether the value occurs i the remaider of the list Backtrackig algorithms A backtrackig algorithm is based o a depth-first recursive search. It Tests to see if a solutio has bee foud, ad if so, returs it; otherwise For each choice that ca be made at this poit, Make that choice Recur If the recursio returs a solutio, retur it If o choices remai, retur failure For example color a map with o more tha four colors: color(coutry ) If all coutries have bee colored ( > umber of coutries) retur success; otherwise, For each color c of four colors, If coutry is ot adjacet to a coutry that has bee colored c Color coutry with color c recursivly color coutry + 1 If successful, retur success Retur failure (if loop exits)

9 Cocept: Amortized aalysis, by Kut Reiert, 18. Oktober 2010, 21: Comparig this to the worst case time of the determiistic Quicksort which is O( 2 ) shows how powerful a coi throw ca be. As a secod cocept i ru time aalysis we have a look at amortized aalysis. Ofte (o matter whether for radomized or determiistic algorithms) the worst case aalysis is ot appropriate, because the wrost case caot happe ofte. No matter what the iput is. If this is the case, the the ru time averaged over a series of operatios caot be equal to times the worst case ru time. Ideed, the averaged ru time will ofte be much better. It is importat to ote that this is differet form the average ru time aalysis. There oe averages over the distributio of iputs. So it could still be that a algorithm, preseted with a bad iput rus very slowly. Here, we average over all possible excutios of the algorithms for ay give iput. To make this distictio clear, this type of aalysis is called amortized aalysis. Cocept: Amortized aalysis Imagie for example a stack. We have the followig operatios o the stack Pop(S) pops the top elemet of the stack ad returs it. Push(S,x) pushed elemet x o the stack. MultiPop(S,k) returs at most the top k elemets from the stack (it calls Pop k times). Obviously the operatios Pop ad Push have worst case time O(1). However the operatio MultiPop ca be liear i the stack size. So if we assume that at most objects are o the stack a multipop operatio ca have worst case cost of O(). Hece i the worst case a series of stack operatios is bouded by O( 2 ). We will ow use this example to illustrate three differet techiques to fid a more realistic amortized boud for operatios. The three methods to coduct the aalysis which are: The aggregate method The accoutig method The potetial method The aggregate method Here we show that for all, a sequece of operatios takes worst-case time T () i total. Hece, for this worst case, the average cost, or amortized cost is T ()/. Note that this method charges the same amortized cost to each operatio i the sequece of operatios, eve if this sequece cotais differet types of operatios. The other two methods ca assig idividual amortized costs for each type of operatio. We argue as follows. I ay sequece of operatios o a iitially empty stack, each object ca be popped at most oce for each time it is pushed. Therefore, the umber of times Pop ca be called o a oempty stack (icludig calls withi MultiPop), is at most the umber of Push operatios, which is at most. Hece, for ay value of, ay sequece of Push, Pop, ad MultiPop operatios takes a total of O() time. Hece the amortized cost of ay operatio is O()/ = O(1). The accoutig method I the accoutig method we assig differig charges to differet operatios, with some operatios charge more or less tha they actually cost. The amout we charge a operatio is called its amortized cost.

10 1010 Cocept: Amortized aalysis, by Kut Reiert, 18. Oktober 2010, 21:22 Whe a operatio s amortized cost exceeds its actual cost, the differece is assiged to specific objects i the data structure as credit. Credit ca the be used to pay later for operatios whose amortized cost is less tha their actual cost. Oe must choose the amortized costs carefully. If we wat the aalysis with amortized costs to show that i the worst case the average cost per operatio is small, the the total amortized cost of a sequece of operatios must be a upper boud o the total actual cost of the sequece. Moreover, this relatioship must hold for all sequeces of operatios, thus the total credit must be oegative at all times. Let us retur to our stack example. The actual costs are: Pop(S) 1, Push(S,x) 1, MultiPop(S,k) mi(k,s), where s is the size of the stack. Let us assig the followig amortized costs. Pop(S) 0, Push(S,x) 2, MultiPop(S,k) 0. Note that the amortized costs of all operatios is O(1) ad hece the amortized cost of operatios is O(). Note also that the actual cost of MultiPop is variable whereas the amortized cost is costat. Usig the same argumet as i the aggregate method it is easy to see that our accout is always charged startig with a empty stack. Each Push operatio pays 2 credits. Oe for its ow cost ad oe for the cost of poppig the elemet off the stack, either through a ormal Pop or through a MultiPop operatio. Please ote that this method ca assig idividual, differet amortized costs to each operatio. The potetial method Istead of represetig prepaid work as credit stored with specific objects i the data structure, the potetial method represets the prepaid work as potetial eergy or simply potetial that ca be released to pay for future operatios. The potetial is associated with the data structure as a whole rather tha with specific objects withi the data structure. It works as follows. We start with a iitial data structure D 0, o which operatios are performed. For each i = 1,2,..., we let c i be the actual cost of the i-th operatio ad D i be the data structure that results after applyig the i-th operatio. A potetial fuctio Φ maps each data structure D i to a real umber Φ(D i ), which is the potetial associated with the data structure D i. The amortized cost ĉ i of the i-th operatio with respect to the potetial fuctio Φ is defied by ĉ i = c i + Φ(D i ) Φ(D i 1 ). that is, the amortized cost is the actual cost plus the icrease i potetial due to its operatio. Hece the total amortized cost of the operatios is: i=1 ĉ i = i=1 (c i + Φ(D i ) Φ(D i 1 )) = i=1 c i + Φ(D ) Φ(D 0 ). If we ca defie a potetial fuctio Φ so that Φ(D ) Φ(D 0 ), the the total amortized cost i=1 ĉi is a upper boud o the total actual cost. I practice we do ot kow how may operatios might be performed ad therefore

11 = k (1.12) Cocept: Amortized aalysis, by Kut Reiert, 18. Oktober 2010, 21: guaratee the Φ(D i ) Φ(D 0 ), for all i. It is ofte coveiet to defie Φ(D 0 ) = 0 ad the show that all other potetials are o egative. Lets illustrate the method usig our stack example. We defie the potetial fuctio o the stack as the umber of elemets it cotais. Hece the empty stack D 0 has Φ(D 0 ) = 0. Sice the umber of objects o the stack is ever egative we have Φ(D i ) 0 for all stacks D i resultig after the i-th operatio. Let us ow compute the amortized costs of the various stack operatios. If the i-th operatio o a stack cotaiig s objects is a Push operatio, the the potetial differece is Φ(D i ) Φ(D i 1 ) = (s + 1) s = 1 (1.11) Hece the amortized cost is: ĉ i = c i + Φ(D i ) Φ(D i 1 ) = = 2. If the i-th operatio o a stack cotaiig s objects is a MultiPop(S,k) the k = mi(k,s) objects are popped off the stack. The actual cost of the operatio is k ad the potetial differece is: Φ(D i ) Φ(D i 1 ) = (s k ) s Hece the amortized cost is: ĉ i = c i + Φ(D i ) Φ(D i 1 ) = k k = 0. A similar result is obtaied for Pop. This shows that the amortized cost of each operatio is O(1) ad hece the total amortized cost of a sequece of operatios is O().

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