Applications of Stacks

Transcription

1 Lists ad Iterators 8// Presetatio for use with the textbook lgorithm esig ad pplicatios, by M. T. Goodrich ad R. Tamassia, Wiley, h ata Structures xkcd Seve Used with permissio uder reative ommos. Licese Stacks First, let us discuss a related structure: the Stack. Isertios ad deletios follow the last-i first-out scheme (LIFO) Thik of a sprig-loaded plate dispeser Mai stack operatios: push(e): iserts a elemet, e pop(): removes ad returs the last iserted elemet uxiliary stack operatios: top(): returs the last iserted elemet without removig it size(): returs the umber of elemets stored isempty(): idicates whether o elemets are stored Example pplicatios of Stacks irect applicatios Page-visited history i a Web browser Udo sequece i a text editor hai of method calls i a laguage supportig recursio Idirect applicatios uxiliary data structure for algorithms ompoet of other data structures Method Stacks The rutime eviromet for such a laguage keeps track of the chai of active methods with a stack Whe a method is called, the system pushes o the stack a frame cotaiig Local variables ad retur value Program couter, keepig track of the statemet beig executed Whe a method eds, its frame is popped from the stack ad cotrol is passed to the method o top of the stack llows for recursio mai() { it i = ; foo(i); } foo(it j) { it k; k = j+; bar(k); } bar(it m) { } bar P = m = foo P = j = k = mai P = i = rray-based Stack simple way of implemetig the Stack T uses a array We add elemets from left to right variable keeps track of the idex of the top elemet S t: umber of items i stack lgorithm size() retur t lgorithm pop() if isempty() the retur ull t t retur S[t] t-

2 Lists ad Iterators 8// rray-based Stack (cot.) The array storig the stack elemets may become full push operatio will the either grow the array or sigal a error S lgorithm push(o) if t = S.legth the sigal stack overflow error t t S[t-] o t- Performace Performace Let be the umber of elemets i the stack The space used is O() Each operatio rus i time O() ualificatios Tryig to push a ew elemet ito a full stack causes a implemetatio-specific exceptio or Pushig a item o a full stack causes the uderlyig array to double i size, which implies each operatio rus i O() amortized time. omputig Spas (ot i book) Usig a stack as a auxiliary data structure i a algorithm Give a array X, the spa S[i] of X[i] is the maximum umber of cosecutive elemets X[j] immediately precedig X[i] ad such that X[j] X[i] Spas have applicatios to fiacial aalysis E.g., stock at -week high X S uadratic lgorithm lgorithm spas(x, ) Iput array X of itegers Output array S of spas of X # S ew array of itegers for i to do s while s i X[i s] X[i] ( ) s s ( ) S[i] s retur S lgorithm spas rus i O( ) time omputig Spas with a Stack We keep i a stack the idices of the elemets larger tha the curret, plus the idex of the curret. We sca the array from left to right Let i be the curret idex We pop idices from the stack util we fid idex j such that X[i] X[j] If stack is empty, we set S[i] i+ otherwise, we set S[i] i j We push i oto the stack S: Stack: Liear Time lgorithm Each idex of the array Is pushed ito the stack exactly oe Is popped from the stack at most oce The body of the while-loop is executed at most times lgorithm spas rus i O() time lgorithm spas(x, ) # S ew array of itegers ew empty stack for i to do while (.isempty() X[.top()] X[i] ) do.pop() if.isempty() the S[i] i S[i] i.top().push(i) retur S

3 Lists ad Iterators 8// ueues I a ueue, isertios ad deletios follow the first-i firstout scheme (FIFO) uxiliary queue operatios: Isertios are at the rear or ed of the queue ad removals are at the frot of size(): returs the umber the queue of elemets stored Mai queue operatios: isempty(): idicates whether o elemets are equeue(e): iserts a elemet, stored e, at the ed of the queue dequeue(): removes ad oudary cases: returs the elemet at the frot of the queue first(): returs the elemet at the frot without removig it ttemptig the executio of dequeue or first o a empty queue sigals a error or returs ull Example Operatio Output equeue() () equeue() (, ) dequeue() () equeue() (, ) dequeue() () first() () dequeue() () dequeue() ull () isempty() true () equeue(9) (9) equeue() (9, ) size() (9, ) equeue() (9,, ) equeue() (9,,, ) dequeue() 9 (,, ) pplicatio: uffered Output The Iteret is desiged to route iformatio i discrete packets, which are at most bytes i legth. y time a video stream is trasmitted o the Iteret, it must be subdivided ito packets ad these packets must each be idividually routed to their destiatio. ecause of vagaries ad errors, the time it takes for these packets to arrive at their destiatio ca be highly variable. Thus, we eed a way of smoothig out these variatios pplicatio: uffered Output This smoothig is typically achieved is by usig a buffer, which is a queue that is used to temporarily store items, as they are beig produced by oe computatioal process ad cosumed by aother. I the case of video packets arrivig via the Iteret, the etworkig process is producig the packets ad the playback process is cosumig them. This producer-cosumer model is eforcig queue, a first-i, first-out (FIFO) protocol for the packets. dditioal pplicatios esides bufferig video, queues also have the followig applicatios: irect applicatios Waitig lists, bureaucracy ccess to shared resources (e.g., priter) Multiprogrammig Idirect applicatios uxiliary data structure for algorithms ompoet of other data structures rray-based ueue Use a array of size N i a circular fashio Two variables keep track of the frot ad size f idex of the frot elemet sz umber of stored elemets Whe the queue has fewer tha N elemets, array locatio r = (f + sz) mod N is the first empty slot past the rear of the queue ormal cofiguratio f r wrapped-aroud cofiguratio r f

4 Lists ad Iterators 8// ueue Operatios We use the modulo operator (remaider of divisio) lgorithm size() retur sz lgorithm isempty() retur (sz ) f r r f ueue Operatios (cot.) Operatio equeue throws a exceptio if the array is full Oe could also grow the uderlyig array by a factor of lgorithm equeue(o) if sz = N the sigal queue full error r (f + sz) mod N [r] o sz (sz + ) f r r f ueue Operatios (cot.) Note that operatio dequeue returs ull if the queue is empty Oe could alteratively sigal a error lgorithm dequeue() if isempty() the retur ull o [f] f (f + ) mod N sz (sz ) retur o f r r f pplicatio: Roud Robi Schedulers We ca implemet a roud robi scheduler usig a queue by repeatedly performig the followig steps:. e =.dequeue(). Service elemet e..equeue(e) ueue equeue Shared Service Equeue Idex-ased Lists idex-based list supports the followig operatios: Example sequece of List operatios:

5 Lists ad Iterators 8// rray-based Lists obvious choice for implemetig the list T is to use a array,, where [i] stores (a referece to) the elemet with idex i. With a represetatio based o a array, the get(i) ad set(i, e) methods are easy to implemet by accessig [i] (assumig i is a legitimate idex). i Isertio I a operatio add(i, o), we eed to make room for the ew elemet by shiftig forward the i elemets [i],, [ ] I the worst case (i ), this takes O() time i i o i Elemet Removal I a operatio remove(i), we eed to fill the hole left by the removed elemet by shiftig backward the i elemets [i ],, [ ] I the worst case (i ), this takes O() time Pseudo-code lgorithms for isertio ad removal: o i i i 8 Performace I a array-based implemetatio of a dyamic list: The space used by the data structure is O() Idexig the elemet at i takes O() time add ad remove ru i O() time i the worst case I a add operatio, whe the array is full, istead of throwig a exceptio, we ca replace the array with a larger oe. Liked Lists Liked lists store elemets at odes or positios. ccessor methods: 9

6 Lists ad Iterators 8// Liked Lists Update methods: Isertio Isert a ew ode, q, betwee p ad its successor. p prev Implemetatio: The most atural way to implemet a positioal list is with a doubly-liked list. ext header odes/positios trailer p q X p q elemet ode Lists ad Iterators elemets X eletio Remove a ode, p, from a doubly-liked list. p Pseudo-code lgorithms for isertio ad deletio i a liked list: p Performace liked list ca perform all of the access ad update operatios for a positioal list i costat time. What is a Tree I computer sciece, a tree is a abstract model of a hierarchical structure tree cosists of odes with a paret-child relatio pplicatios: Orgaizatio charts File systems Programmig eviromets US Sales Iteratioal omputers R Us Maufacturig Laptops Europe sia aada esktops R&

7 Lists ad Iterators 8// Tree Termiology Root: ode without paret () Subtree: tree cosistig of Iteral ode: ode with at least a ode ad its oe child (,,, F) descedats Exteral ode (a.k.a. leaf ): ode without childre (E, I, J, K, G, H, ) cestors of a ode: paret, gradparet, grad-gradparet, etc. epth of a ode: umber of acestors E F G H Height of a tree: maximum depth of ay ode () escedat of a ode: child, gradchild, grad-gradchild, etc. I J K subtree Tree Operatios ccessor methods: uery methods: Geeric methods: 8 Preorder Traversal traversal visits the odes of a tree i a systematic maer I a preorder traversal, a ode is visited before its descedats pplicatio: prit a structured documet Make Moey Fast! 9. Motivatios. Methods Refereces. Greed. vidity 8. Stock Fraud lgorithm preorder(v) visit(v) for each child w of v preorder (w). Pozi Scheme. ak Robbery 9 Postorder Traversal I a postorder traversal, a ode is visited after its descedats pplicatio: compute space used by files i a directory ad its subdirectories 9 cs/ hc.doc K homeworks/ hc.doc K R.java K lgorithm postorder(v) for each child w of v postorder (w) visit(v) programs/ Stocks.java K Robot.java K 8 todo.txt K iary Trees rithmetic Expressio Tree biary tree is a tree with the followig properties: Each iteral ode has at most two childre (exactly two for proper biary trees) The childre of a ode are a ordered pair We call the childre of a iteral ode left child ad right child lterative recursive defiitio: a biary tree is either a tree cosistig of a sigle ode, or a tree whose root has a ordered pair of childre, each of which is a biary tree pplicatios: arithmetic expressios decisio processes searchig E F G H I iary tree associated with a arithmetic expressio iteral odes: operators exteral odes: operads Example: arithmetic expressio tree for the expressio ( (a ) ( b)) a b

8 Lists ad Iterators 8// ecisio Tree Properties of Proper iary Trees iary tree associated with a decisio process iteral odes: questios with yes/o aswer exteral odes: decisios Example: diig decisio Wat a fast meal? Yes No How about coffee? O expese accout? Yes No Yes No Notatio umber of odes e umber of exteral odes i umber of iteral odes h height Properties: e i e h i h ( ) e h h log e h log ( ) Starbucks hipotle Gracie s afé Parago iary Tree Operatios biary tree exteds the Tree operatios, i.e., it iherits all the methods of atree. dditioal methods: positio lefthild(v) positio righthild(v) positio siblig(v) The above methods retur ull whe there is o left, right, or siblig of p, respectively Update methods may be defied by data structures implemetig the biary tree Iorder Traversal I a iorder traversal a ode is visited after its left subtree ad before its right subtree pplicatio: draw a biary tree x(v) = iorder rak of v y(v) = depth of v lgorithm iorder(v) if left (v) ull iorder (left (v)) visit(v) if right(v) ull iorder (right (v)) 8 9 Prit rithmetic Expressios Specializatio of a iorder traversal prit operad or operator whe visitig ode prit ( before traversig left subtree prit ) after traversig right subtree a b lgorithm pritexpressio(v) if left (v) ull prit( ( ) iorder (left(v)) prit(v.elemet ()) if right(v) ull iorder (right(v)) prit ( ) ) (( (a )) ( b)) Evaluate rithmetic Expressios Specializatio of a postorder traversal recursive method returig the value of a subtree whe visitig a iteral ode, combie the values of the subtrees lgorithm evalexpr(v) if isexteral (v) retur v.elemet () x evalexpr(left(v)) y evalexpr(right(v)) operator stored at v retur x y 8 8

9 Lists ad Iterators 8// Euler Tour Traversal Geeric traversal of a tree Travel each edge exactly twice. L R Liked Structure for Trees ode is represeted by a object storig Elemet Paret ode Sequece of childre odes Node objects implemet the Positio T F F 9 E E Liked Structure for iary Trees ode is represeted by a object storig Elemet Paret ode Left child ode Right child ode Node objects implemet the Positio T E E rray-ased Represetatio of iary Trees Nodes are stored i a array G H 9 Node v is stored at [rak(v)] rak(root) = if ode is the left child of paret(ode), E F J rak(ode) = rak(paret(ode)) + if ode is the right child of paret(ode), 9 rak(ode) = rak(paret(ode)) G H 9