Andersen s Algorithm. CS 701 Final Exam (Reminder) Friday, December 12, 4:00 6:00 P.M., 1289 Computer Science.

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1 CS 701 Finl Exm (Reminde) Fidy, Deeme 12, 4:00 6:00 P.M., 1289 Comute Siene. Andesen s Algoithm An lgoithm to uild oints-to gh fo C ogm is esented in: Pogm Anlysis nd Seiliztion fo the C ogmming Lnguge, L.O. Andesen, The lgoithm exmines sttements tht ete ointes, one y one, in textul ode (the lgoithm is flowinsensitive). Eh sttement udtes the oints-to gh if it n ete new oints-to eltionshis Six kinds of sttements e onsideed: = &; = q; = *; * = &; * = q; * = *; 2. = q; We dd s fom to eveything q oints to. If new s fom q e lte dded, oesonding s fom must lso e dded (this imlies n itetive o woklist lgoithm). Fo exmle (the dshed is newly dded): We will detil the oints-to gh udtes eh of the sttements indues. 1. = &; We dd n fom to, showing n ossily oint to : q

2 3. = *; Let S e ll the nodes oints to. Let T e ll the nodes memes of S oint to. We dd s fom to ll nodes in T. If lte ointe ssignments inese S o T, new s fom must lso e dded (this gin imlies n itetive o woklist lgoithm). Fo exmle (dshed s e newly dded): d e f 4. * = &; Add n to fom ll nodes oints to. If new s fom e lte dded, new s to must e dded (this imlies n itetive o woklist lgoithm). Fo exmle (dshed s e newly dded): q * = q; Nodes ointed to y must e linked to ll nodes ointed to y q. If lte ointe ssignments dd s fom o q, this ssignment must e evisited (this gin imlies n itetive o woklist lgoithm). Fo exmle (dshed s e newly dded): e 6. * = *; Let S e ll the nodes oints to. Let T e ll the nodes memes of S oint to. We dd s fom ll nodes oints to to ll nodes in T. If lte ointe ssignments inese S o T o link new nodes to, this ssignment must e evisited (this gin imlies n itetive o woklist lgoithm). Fo exmle (dshed s e newly dded): s f q u v d w x

3 Exmle Next: Conside the following ointe mniultions: 1 = &; 2 = &; 1 = 2; = &1; * = &; 3 = *; 2 = &d; We stt with: 1 = &; 2 = &; 1 1 = 2; 1 2 Then: = &1; Next: Finlly: * = &; 2 = &d; Then: 3 = *; d But we en t quite done yet. This lgoithm is flow-insensitive, so we must onside othe exeution odes (nd itetive e-exeution). If we mke nothe ss though the ssignments, we see tht the finl

4 ssignment to 2 n flow to 1, nd then to 3 though : 1 2 d 3 This oints-to gh is the dense, ut it does tue ll the wys ointe vlues might ogte though the vious ointe ssignments. Clls e hndled y teting ointe metes nd ointe etuns s ssignments, done t the oints of ll nd etun. Suogm odies e effetively inlined to tue the oints-to eltions they indue. Given *int eho (*int ) { etun ; } = eho (&); we see the imliit ssignments = &; = ; nd dd the following oints-to infomtion: As n otimiztion, liies n e e-nlyzed to detemine the oints-to eltions they indue. Most my use (ed) ointes ut don t ete ny new oints-to eltions visile outside thei odies. Cll to suh liy outines n e ignoed s f s the lle s oints-to gh is onened. Pefomne of Andesen s Algoithm Exeiene hs shown tht Andesen s Algoithm gives useful oints-to dt nd is f sueio to the nive ddess-tken oh. Inteestingly, exeiments show tht mking the tehnique flow-sensitive o lling ontext-sensitive doesn t imove esults vey muh on tyil enhmks. But exeution time fo modete to lge ogms n e olem. Ceful nlysis shows tht Andesen s Algoithm n equie O(n 3 ) time (whee n is the nume of nodes in the oints-to gh)

5 The eson fo this lge-thnexeted nlysis time is tht sttement like = *q; n foe the lgoithm to visit n 2 nodes (q my oint to n nodes nd eh of these nodes my oint to n nodes). The nume of ointe sttements nlyzed n e O(n), leding to n O(n 3 ) exeution time. Steensgd s Algoithm It would e useful to hve esonly ute oints-to nlysis tht uns in essentilly line time so tht elly lge ogms ould e hndled. This is wht Steensgd s Algoithm offes. (Points-to Anlysis in Almost Line Time, B. Steensgd, 1996 Piniles of Pogmming Lnguges Confeene.) Steensgd s Algoithm is essentilly Andesen s Algoithm, simlified y meging nodes nd if ny ointe n efeene oth Tht is, in Andesen s Algoithm we might hve Steensgd s Algoithm is sometimes less ute thn Andesen s Algoithm. Fo exmle, the following oints-to gh, eted y Andesen s Algoithm, shows tht my oint to o whees q my only oint to : In Steensgd s Algoithm we would insted hve q In effet ny two lotions tht might e ointed to y the sme ointe e led in single equivlene lss. In Steensgd s Algoithm we get q inoetly showing tht if my oint to o then so my q

6 But now sttements like = *q; n t foe the lgoithm to visit n 2 nodes, euse multile nodes efeened y the sme ointe e lwys meged. Using the fst unionfind lgoithm, we n get n exeution time of O(n α(n)) whih is essentilly line in n. Now vey lge ogms n e nlyzed, nd without too muh of loss in eision. Andesen vs. Steensgd in Ptie Howitz nd Shio exmined 61 C ogms, nging in size fom 300 to 24,300 lines. As exeted, Steensgd is less eise: On vege oints-to sets e 4 times igge; t wost 15 times igge. As exeted, Andesen is slowe. On vege 1.5 times slowe: t wost 31 times slowe. Both e muh ette thn the nive ddess tken oh. Bottom line: Use Andesen fo smll ogms, use Steensgd (o something else) fo lge ogms The Howitz-Shio Aoh It would e nie to hve oints-to nlysis tht is meteizle, nging etween the uy of Andesen nd the seed of Steensgd. Howitz nd Shio (Fst nd Aute Flow-Insensitive Points-To Anlysis, 1997 Piniles of Pogmming Lnguges Confeene) esent tehnique intemedite to those oosed y Andesen nd Steensgd. Howitz nd Shio suggest eh node in the oints-to gh e limited to out degee k, whee 1 k n. If k =1 then they hve Steensgd s oh. If k =n (n is nume of nodes in oints to gh), then they hve Andesen s oh. Thei wost se un-time is O(k 2 n), whih is not muh wose thn Steensgd if k is ket esonly smll

7 To use thei oh ssign eh vile tht my e ointed to to one of k tegoies. Now if my oint to x nd my lso oint to y, we mege x nd y only if they oth e in the sme tegoy. If x nd y e in diffeent tegoies, they en t meged, leding to moe ute oints-to estimtes. Exmle 1 = &; 1 = &; 1 = &; 2 = &; Sy we hve k = 2 nd le nd in tegoy 1 nd in tegoy 2. We then uild: 1, 2 This oints-to gh is just s ute s tht uilt y Andesen s oh But... Wht if we hose to le in tegoy 1 nd nd in tegoy 2. We now hve: 1 2, This gh is inext, sine it tells us 2 my oint to, whih is flse. (Steensgd would hve een wose still, inoetly telling us 2 my oint to s well s nd ). Anothe Good Ide Wht if we n Shio nd Howitz s oints-to nlysis twie, eh with diffeent tegoy ssignments? Eh un my odue diffeent oints-to gh. One my sy 2 oints to whees the othe sys it n t. Whih do we elieve? Neithe nlysis misses genuine oints-to eltion. Rthe, meging of nodes sometimes etes flse ointsto infomtion. So we will elieve 2 my oint to only if ll uns sy so. This mens multile uns my filte out mny of the flse oints-to eltions used y meging

8 How Mny Runs e Needed? How e Ctegoies to e Set? We wnt to ssign tegoies so tht duing t lest one un, ny i of ointed-to viles e in diffeent tegoies. This guntees tht if ll the uns tell us my oint to nd, it is not just euse nd lwys hened to e ssigned the sme tegoy. To foe diffeent tegoy ssignments fo eh i of viles, we ssign eh ointed-to vile n index nd wite tht index in se k (the nume of tegoies hosen). Fo exmle, if we hd viles,, nd d, nd hose k = 2, we d use the following iny indies: d 11 Note tht the nume of se k digits needed to eesent indies fom 0 to n-1 is just eiling(log k n). This nume is just the nume of uns we need! Why? In the fist un, we ll use the ight most digit in vile s index s its tegoy. In the next un, we ll use the seond digit fom the ight, then the thid digit fom the ight,... Any two distint viles hve diffeent index vlues, so they must diffe in t lest digit osition. Retuning to ou exmle, d 11 On un #1 we give nd tegoy 0 nd nd d tegoy 1. On un #2, nd get tegoy 0 nd nd d get tegoy 1. So using just 2 uns in this simle se, we eliminte muh of the inuy Steensgd s meging intodues. Run time is now O(log k (n) k 2 n)

9 How Well does this Aoh Wok? On 25 tests, using 3 tegoies, Howitz & Shio oints-to sets on vege e 2.67 lge thn those of Andesen (Steensgd s e 4.75 lge). This oh is slowe thn Steensgd ut on lge ogms it is 7 to 25 times fste thn Andesen. How Well do Points-to Anlyses Wok in Rel Dt Flow Polems? In Whih Pointe Anlysis Should I Use, Hind nd Pioli suvey the effetiveness of nume of ointsto nlyses in tul dt flow nlyses (mod/ef, liveness, ehing defs, inteoedul onstnt ogtion). Thei onlusions e essentilly the sme oss ll these nlyses: Steensgd s nlysis is signifintly moe eise thn ddess-tken nlysis nd not signifintly slowe. Andesen s nlysis odues modest, ut onsistent, imovements ove Steensgd s nlysis Both ontext-sensitive oints-to nlysis nd flow-sensitive oints-to nlysis give little imovement ove Andesen s nlysis. 615

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