Problem Final Exam Set 2 Solutions

Transcription

1 CSE 5 5 Algoritms nd nd Progrms Prolem Finl Exm Set Solutions Jontn Turner Exm - //05 0/8/0. (5 points) Suppose you re implementing grp lgoritm tt uses ep s one of its primry dt strutures. Te lgoritm does t lest n insert nd n deletemin opertions nd t most m / insert nd m / deletemin opertions, wen m / >n. It lso does t most m ngekey opertions, ll of wi redue te key vlue. If you implemented tis lgoritm using d-eps, wt vlue of d would you use to get te lowest symptoti running time. (Hint: rek tis into two ses; one for smller vlues of m, one for lrger vlues.) Wt is te resulting symptoti running time? Wt is te resulting symptoti running time if you used Fioni ep, insted of d- ep? Wi type of ep would you oose for tis pplition? Wy? - -

2 . (5 points) In te Fioni eps dt struture, ut etween vertex u nd its prent v uses sding ut t v if v s lredy lost ild sine it lst eme ild of some oter vertex. Suppose we nge tis, so tt sding ut is performed t v only if v s lredy lost two ildren. How does tis nge lter te lemm sown elow (tis lemm is from te nlysis of te running time of Fioni eps)? Explin your nswer. Lemm. Let x e ny node in n F-ep. Let y,...,y r e te ildren of x, in order of time in wi tey were linked to x (erliest to ltest). Ten, rnk(y i ) i for ll i. Let S k e te smllest possile numer of desendnts tt node of rnk k s, in our modified version of Fioni eps. Give reursive lower ound on S k. Tt is, give n inequlity of te form S k f(s 0,S,..., S k ) were f is some funtion of te S i s for i<k. Use tis to give lower ound on te smllest numer of desendnts tt node wit rnk n ve. - -

3 . (5 points) Suppose you re sked to write progrm tt responds to series of queries out given grp of te form Will te grp ve negtive yle, if te weigt of edge (u,v) is nged to x? You my ssume tt te grp s no negtive yles, ltoug it does ve negtive edges. Desrie (in words) n lgoritm tt responds to su queries in O(mn log n) time. Your lgoritm my pre-ompute ertin informtion efore te first query is reeived, ut te mount of dditionl informtion is limited to O(n) new vlues. How mu time does your lgoritm need to initilize tis extr informtion? Sow te uxiliry informtion your metod would ompute for te grp sown elow. d e f g - -

4 . (0 points). Te figure elow sows n intermedite stte in te exeution of Dijkstr s lgoritm. Te old edges in te grp re te edges defined y te prent pointers, nd te numers next to te verties re te urrent distne vlues. Fill in te lnks (s pproprite) in te rrys tt implement te d-ep (ssume d=) e d f i g j k key d e f g i j k l Sow ow te ep ontent nges fter te next itertion. key d e f g i j k l - -

5 5. (5 points) Consider inry ser tree in wi e vertex s n ssoited key nd ost. Te verties re ordered y te keys in te usul wy (so te keys of te verties in te left sutree of given vertex x re stritly less tn te key of x, nd so fort). Te osts re represented using te differentil representtion we used for representing pt sets. Complete te reursive funtion ser, sown elow, so tt it returns te node wit te smllest key from mong tose verties wit osts less tn or equl to given ound. Te struture of te tree nodes is sown elow lso. lss twowytrees { int n; // trees defined on items {,...,n} strut node { int k, D, Dm; // key nd differentil ost fields int l, r; // indies of left nd rigt ildren } *ve;... } #define left(x) (ve[x].l) // you my ssume similr delrtions // for rigt, key, Dost, Dmin int ser(int t, int ostbound, int dmsum) { // Return te index of te leftmost node in te sutree wit // root t tt s ost <= ostbound. Te vrile dmsum is // te sum of te Dmin vlues for te proper nestors of t. // Return Null, if tere is no node wit ost less tn ostound. } - 5 -

6 6. (5 points) In tis prolem, you re to sow tt te generl preflow-pus lgoritm tkes O(mn) time to find mximum flow in grp in wi ll edges ve pity. (Rell tt te ound for generl grps is O(mn ).) How mny steps dd flow to n edge in te residul grp witout sturting it? Wy? Explin wy te time spent on releling steps is O(mn). Explin wy te time spent on steps tt sturte n edge in te residul grp is O(mn). Explin wy te time spent finding dmissile edges (using te nextedge pointers) is O(mn)

7 - -. (5 points) Given iprtite, grp wit edge weigts. Drw piture of te min-ost flow grp tt n e used to find mximum mting in tis grp. x z y 8 6 x z y 8 6

8 Identify minimum ost ugmenting pt in te flow grp. Wt is its ost? Drw te residul grp tt results from sturting tis pt. Identify minimum ost ugmenting pt in te residul grp. Wt is its ost? Wt is te mting orresponding to te flow tt exists fter flow is dded to tis ugmenting pt? - 8 -

9 8. (0 points) Te figure elow sow possile intermedite stge in te exeution of te Edmonds-Krp lgoritm for finding mximum size mting. Te sttes of te verties re indited y te plus nd minus signs (for even nd odd), te rrows represent te prent pointers nd te edges djent to ertin verties orrespond to teir ridge vlues. Te prtition dt struture is sown t rigt nd te queue of edges to e proessed is t te top. In te digrm t left, drw losed urve round te sets of verties tt ve een ondensed into single vertex in te urrent srunken grp. If ny of tese sets ontin susets, orresponding to smller lossoms, irle te verties in tose smller lossoms, s well. queue: e, i, l, kd, ln g d e r j ln k g j pq q e p d g l g qp p n s m n m s i q k l i r Wt ppens wen te next edge in te queue is proessed y te lgoritm? Explin. Wen te next edge, [i,] is proessed, n ugmenting pt is found. Wt is tt ugmenting pt? - 9 -