## Lower And Upper Bounds For Incremental Algorithms (1992)

Citations: | 3 - 0 self |

### BibTeX

@TECHREPORT{Berman92lowerand,

author = {Arthur Michael Berman},

title = {Lower And Upper Bounds For Incremental Algorithms},

institution = {},

year = {1992}

}

### OpenURL

### Abstract

An incremental algorithm (also called a dynamic update algorithm) updates the answer to some problem after an incremental change is made in the input. We examine methods for bounding the performance of such algorithms. First, quite general but relatively weak bounds are considered, along with a careful examination of the conditions under which they hold. Next, a more powerful proof method, the Incremental Relative Lower Bound is presented, along with its application to a number of important problems. We then examine an alternative approach, delta-analysis, which had been proposed previously, apply it to several new problems and show how it can be extended. For the specific problem of updating the transitive closure of an acyclic digraph, we present the first known incremental algorithm that is efficient in the delta-analysis sense. Finally, we criti...

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Citation Context ...nd the speed of incremental algorithms for this problem. 3-SAT is the problem of determining whether a set of clauses, each clause containing exactly three literals, has a satisfying truth assignment =-=[GJ79]-=-. We will show that no NP-complete problem can have a polynomial time incremental algorithm (with restricted history) unless P = NP . Theorem 3.4.1 There does not exist an incremental algorithm (with ... |

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Citation Context ...l, direct analysis of the complexity of the algorithm, which is not provided by Ramalingam and Reps, might yield a better bound. However, since the algorithm is based directly on Dijkstra's algorithm =-=[Dij59]-=- for shortest paths, it is certainly not going to be any better than O(n 3 ). The point of the above analysis is not to question the utility of the Ramalingam and Reps algorithm --- when ffi is small ... |

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Citation Context ... efficient, or simpler, than a fully-dynamic incremental algorithm. For example, the method of Even and Shiloach for updating connected 5 List maintenance problems are a notable exception; see, e.g., =-=[ST85]. 94 compo-=-nents requires O(n) per deletion (amortized) [ES81]. By working in only one direction, a partially-dynamic algorithm can "throw away" structure as it goes, whereas a fully-dynamic algorithm ... |

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Citation Context ...nd connected components problems can be modeled as well. For an overview of data flow analysis problems, see [ASU86]; for a discussion of the system equations model applied to data flow analysis, see =-=[Hec77]-=- and [RP86]. A system of equations of form (4.5.1), can be modeled as a digraph constructed from the variable interdependencies. Each variable corresponds to a node in the digraph; the appearance of X... |

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Citation Context ...panning Trees Frederickson has presented an algorithm for incremental update of edges in a Minimal Spanning Tree (MST), and an extension that can be used to update the Connected Components in a graph =-=[Fre85]-=-. His algorithm runs in time O( p m), where m is the 35 number of edges in the graph. The algorithm requires O(m) preprocessing, plus the time to find the initial minimum spanning tree, which is known... |

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Citation Context ...onents [Ita91, BT90], -- transitive closure [IK83, Ita86, Ita88, LPv88, YS88, Yel91], -- planar graphs [Tam88, TP90, BT89, EIT + 92, PT88]; ffl computational geometry [Ov81, CBT + 92]; ffl data bases =-=[ABJ89]-=-; ffl syntax-directed editors and grammars [Rep82, RTD83, Rep88, ACR + 87]; ffl data-flow analysis [Ryd83, RP88, RC86, CR88, Mar89, Bur90, PS89, Zad84, Gho83, KRvM88], ffl code generation and optimiza... |

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Citation Context ...e O( p m), where m is the 35 number of edges in the graph. The algorithm requires O(m) preprocessing, plus the time to find the initial minimum spanning tree, which is known to be O(m log fi(m; n)) 6 =-=[GGST86]-=-. In Section 4.4.2, we described a method for demonstrating that MST is in IRLB Class 2. Implicit in this proof is that Step 5 in the IRLB proof procedure (Figure 4.1), which corresponds to the prepro... |

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78 | Incremental planarity testing - Battista, Tamassia - 1989 |

65 | Incremental Algorithms for Minimal Length Paths - Ausiello, Italiano, et al. - 1990 |

53 | Elimination algorithms for data flow analysis
- Ryder, Paull
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(Show Context)
Citation Context ... components problems can be modeled as well. For an overview of data flow analysis problems, see [ASU86]; for a discussion of the system equations model applied to data flow analysis, see [Hec77] and =-=[RP86]-=-. A system of equations of form (4.5.1), can be modeled as a digraph constructed from the variable interdependencies. Each variable corresponds to a node in the digraph; the appearance of X j on the r... |

46 |
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(Show Context)
Citation Context ...IRLB's better than 1=`. Some of the arguments appear in this paper; the others can be derived easily. Problems with O(1) IRLB's (Class 3) Reference Shortest path in a digraph (various forms) x 4.4.1, =-=[AHU74]-=- Transitive Closure of a binary relation x 6.3 Planarity Testing [AHU74] Strong Connectivity [AHU74] Minimum-Maximum edge weight path x 4.5.1, [Pau88] Reaching Definitions x 4.5.2, [RP86] Available Ex... |

45 | Incremental data flow analysis algorithms - Ryder, Paull - 1988 |

43 | An incremental version of iterative data flow analysis - Pollock, Soffa - 1989 |

42 |
Data Structures and Algorithms 2: Graph Algorithms and NP-Completeness
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(Show Context)
Citation Context ...clic graph or DAG. Transitive closure on a DAG can be computed in O(njE red j). E red is the set of edges in the reduced graph for G, that is, the smallest graph with the same transitive closure as G =-=[Meh84]-=-. In the worst case, E red = O(n 2 ), so the DAG algorithm requires O(n 3 ); as a practical matter it is much faster than the standard multiplicative algorithm for 51 many graphs. 6.2 Incremental Uppe... |

41 | Incremental Evaluation for Attribute Grammars with Application to Syntax-directed Editors - Demers, Reps, et al. - 1981 |

39 | Data Structures and Their Algorithms - Lewis, Denenberg - 1991 |

39 | Leeuwen. Maintenance of transitive closure and transitive reduction of graphs
- PoutrĂ©, van
(Show Context)
Citation Context ... case. This improves on Ibaraki and Katoh's result when m AE n. Again, it is possible that in the worst case a single deletion could require O(nm) time. Similar results were reported independently in =-=[LPv88]-=-, who also extend their deletion method to general digraphs, at a higher time complexity. More recently, Yellin has improved on these when the degree of the graph is bounded [Yel91]. His approach hand... |

37 | Maintaining bridge-connected and biconnected components on-line - Westbrook, Tarjan - 1992 |

34 |
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Citation Context ... worst case performance could be quite bad, and the amortized performance no better since it would be possible to get in a situation where the worst case was repeated an arbitrary number of times. In =-=[Ita86]-=- Italiano introduced an algorithm that can process a sequence of n edge additions and searchpath operations in O(n) amortized time per operation. Hence a sequence of q = O(m) edge additions requires O... |

32 |
On-line computation of transitive closure of graphs
- Ibaraki, Katoh
- 1983
(Show Context)
Citation Context ...rd multiplicative algorithm for 51 many graphs. 6.2 Incremental Upper Bounds There has been quite a bit of work in developing incremental algorithms for this problem, beginning with Ibaraki and Katoh =-=[IK83]-=-. 1 Ibaraki and Katoh give algorithms for updating the transitive closure for a graph with edge additions and edge deletions. After each change in the graph, the transitive closure information can be ... |

32 |
Finding paths and deleting edges in directed acyclic graphs
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- 1988
(Show Context)
Citation Context ...uery pathexists(a,b) in O(1) time. A single addition (or searchpath) can require O(nm) time in the worst case. Deletions are not addressed. Italiano has also developed algorithms for updates on DAG's =-=[Ita88]-=-. These algorithms support a sequence of edge deletions in O(n) amortized time per deletion. A sequence of any number of deletionssm requires a total of O(nm) in the worst case. This improves on Ibara... |

32 | Incremental data flow analysis in a structure program editor - Zadeck - 1984 |

30 |
Incremental Data Flow Analysis via Dominator and Attribute Update
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- 1988
(Show Context)
Citation Context ...a91], large clusters of vertices in the graph are condensed into clusters, and links are maintained between the clusters. In Carroll and Ryder's algorithms for updating attribute and dominator graphs =-=[CR88]-=-, a tree is laid over the target graph, and pointers can be followed along the tree edges, which are not in general the same as edges in the graph. Thus, lower bounds in the local persistence model, w... |

28 |
INC: A Language for Incremental Computations
- Yellin, Strom
- 1991
(Show Context)
Citation Context ...eterrent to the value of the bound in most cases. First, many of the problems for which there has been interest in incremental methods, such as Connected Components and Transitive Closure (see, e.g., =-=[YS88]-=-), have linear time start-over algorithms; in this case, the lower bound is also known (linear) and hence an 7 The reader may recognize that the signature corresponds to the upper-diagonal portion of ... |

26 |
Optimal-time incremental semantic analysis for syntax-directed editors
- REPS
(Show Context)
Citation Context ...s ffi-analysis, and algorithms that are efficient in terms of ffi-analysis are referred to as ffi -incremental algorithms. 1 This is a generalization of an idea that first appeared in a paper by Reps =-=[Rep82]-=- and was extended by Alpern et al. [AHR + 90]. Reps presented an incremental algorithm for updating an attribute graph with performance linear in the number of vertices in the graph whose attributes a... |

24 |
An on-line edge deletion problem
- Even, Shiloach
- 1981
(Show Context)
Citation Context ...rithm. For example, the method of Even and Shiloach for updating connected 5 List maintenance problems are a notable exception; see, e.g., [ST85]. 94 components requires O(n) per deletion (amortized) =-=[ES81]. By worki-=-ng in only one direction, a partially-dynamic algorithm can "throw away" structure as it goes, whereas a fully-dynamic algorithm typically needs to maintain complicated history information. ... |

23 |
A critical analysis of incremental iterative data flow analysis algorithms
- Burke, Ryder
- 1990
(Show Context)
Citation Context ... for average case complexity. For some domains, it may be possible to compare the utility of incremental algorithms considering their performance for the types of inputs that are most likely to occur =-=[BR90]-=-. There have also been a few attempts to demonstrate the utility of incremental algorithms by comparing their performance on random data, versus starting over from scratch; see [Car88, RLP90]. While i... |

23 |
Speeding up dynamic transitive closure for bounded degree graphs
- Yellin
- 1993
(Show Context)
Citation Context ...d independently in [LPv88], who also extend their deletion method to general digraphs, at a higher time complexity. More recently, Yellin has improved on these when the degree of the graph is bounded =-=[Yel91]-=-. His approach handles a sequence of m insertions in time O(dm 3 ), where d is the degree bound and m 3 is the size of the transitive closure. Thus the cost per operation is O( dm 3 m ), which, depend... |

22 | On-line graph algorithms with SPQR-trees - Battista, Tamassia - 1990 |

19 |
On the Computational Complexity of Incremental Algorithms
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- 1991
(Show Context)
Citation Context ...r than an algorithm that solves the problem from scratch [EG85]. This work inspired the developments described in Chapter 4. Work begun by Alpern et al. [AHR + 90] and extended by Reps and Ramalingam =-=[RR91]-=- gives lower bounds based on ffi-analysis, within a limited model of computation called local persistence; their work is described and extended further in Chapter 5. The contributions of this disserta... |

18 | On-line algorithms for polynomially solvable satisfiability problems - Ausiello, Italiano - 1991 |

17 | Incremental data flow analysis - Ryder - 1983 |

13 | Persistent data structures - Sarnak - 1986 |

12 |
Proving relative lower bounds for incremental algorithms
- Berman, Paull, et al.
- 1990
(Show Context)
Citation Context ... solve the connected components problem, which was shown in Section 4.4.2 to be in Class 2. It can be shown that any problem that can be modeled in this way is in Class 2; the details can be found in =-=[BPR86]-=-. 4.6 Implications 4.6.1 A Collection of IRLB's Figure 4.7 lists certain common functions for which we have proven IRLB's better than 1=`. Some of the arguments appear in this paper; the others can be... |

12 | Algorithms for updating minimum spanning trees - Chin, Houck - 1978 |

12 | Data Flow Analysis and Incremental Iteration - Marlowe - 1989 |

12 | An incremental algorithm for software analysis - Carroll, Ryder - 1987 |

11 |
A Topological Approach to Dynamic Graph Connectivity
- Reif
- 1987
(Show Context)
Citation Context ... This method is used to show that the Single-Source Reachability problem is non-ffi-incremental for locally persistent algorithms. The second method is a problem reduction approach similar to that in =-=[Rei87]-=-. This method is used to extend the reachability result to Single-Source (or Single-Sink) Closed-Semiring Path problems and Meet-Semilattice Data-Flow Analysis problems. 5.4 New ffi-Analysis Lower Bou... |

11 | The parametric problem of shortest distances - Rodionov - 1968 |

10 | A new shortest path updating algorithm - Goto, Sangiovanni-Vincentelli - 1978 |

10 | A dynamic data structure for planar graph embedding - Tamassia - 1988 |