Results

**1 - 10**of**10**### Table 1 Main results for IMFP, IMCP and their subproblems IMFP IMCP UnSplitFlow CapPath EdgeDisjPath Multiterminal

2001

"... In PAGE 13: ....-C. Costa et al. / European Journal tiflow problems, and of their subproblems, is often in planar graphs, the multiterminal cut and inte- gral flow problems in acyclic graphs and the multicut and integral flow problems in directed trees. Finally, we would like to point out that there are still some interesting opened questions: what is the complexity of the two flow problem in directed bipartite graphs? Is it possible to approximate within a constant ratio the minimum multicut problem in unrestricted graphs? To conclude this survey, we give Table1 which summarizes the most significant results. References [1] R.... ..."

### Table 1. Description of test circuits Test

"... In PAGE 5: ...anguage. We compare it with SSTT [7]. For fair comparison, both programs are compiled and run on a SUN V880 workstation. MCNC and IBM benchmarks are used as the test cases in Table1 . Besides, we randomly generate two more test cases, by setting the percentage of multi-terminal nets and large nets.... ..."

Cited by 1

### Table 1. Performance of AI_CMP for some Benchmark channels

"... In PAGE 4: ...how that on an average, overall reduction in crosstalk is 37.5% for AI_CMP. Figure 6 shows the variation of crosstalk with increasing channel length for LEA, a TSP-based approach [21] and AI_CMP. Results for some benchmark channels are shown in Table1 . Multi-terminal nets were considered as two-terminal nets with the two terminals at extremities.... ..."

### Table 3: Wirelength Reduction 2-term Nets M-term Nets Total

"... In PAGE 4: ... We select 15 designs from our regression tests to cover different type of designs. We show the wirelength difference in Table3 . Note that, we keep the same placement for both 2-geometry routing and 4-geometry routing.... In PAGE 4: ... For the multiterminal nets, we have an average of 6.16% in column 3 of Table3 . The total wirelength re- duction is 6.... ..."

### Table 1.Gain of a Candidate Block According to a Single Net pin in K ! status in-pin edge output total

2001

"... In PAGE 4: ... Actually, by adding block B to cluster C, an input pin of the cluster gets free and can be used for another net connection. In Table1 this is de ned as in-pin gain. This increases the probability of acceptance of adding the block to the cluster.... In PAGE 4: ... This means that there would be no connection from outside to this pin since all the terminals of the corresponding net is located inside the cluster. Therefore the number of external connections of the cluster decreases, de ned as output congestion gain in Table1 . This yields less congestion among the clusters.... In PAGE 4: ... Therefore, reduc- ing an edge from a multi-terminal net should not be considered equivalent to reducing an edge resulting from inserting a two- terminal net inside a cluster. The average gain that a block can take from an n-terminal net i connected to one of its pins, depend- ing on type of the net can be estimated from Table1... ..."

Cited by 6

### Table 1.Gain of a Candidate Block According to a Single Net pin in K ! status in-pin edge output total

2001

"... In PAGE 4: ... Actually, by adding block B to cluster C, an input pin of the cluster gets free and can be used for another net connection. In Table1 this is de ned as in-pin gain. This increases the probability of acceptance of adding the block to the cluster.... In PAGE 4: ... This means that there would be no connection from outside to this pin since all the terminals of the corresponding net is located inside the cluster. Therefore the number of external connections of the cluster decreases, de ned as output congestion gain in Table1 . This yields less congestion among the clusters.... In PAGE 4: ... Therefore, reduc- ing an edge from a multi-terminal net should not be considered equivalent to reducing an edge resulting from inserting a two- terminal net inside a cluster. The average gain that a block can take from an n-terminal net i connected to one of its pins, depend- ing on type of the net can be estimated from Table1... ..."

Cited by 6

### Table 1.Gain of a Candidate Block According to a Single Net

2001

"... In PAGE 4: ... Actually,by adding blockBto cluster C, an input pin of the cluster gets free and can be used for another net connection. In Table1 this is de ned as in-pin gain. This increases the probability of acceptance of adding the blockto the cluster.... In PAGE 4: ... This means that there would be no connection from outside to this pin since all the terminals of the corresponding net is located inside the cluster. Therefore the number of external connections of the cluster decreases, de ned as output congestion gain in Table1 . This yields less congestion among the clusters.... In PAGE 4: ... Therefore, reduc- ing an edge from a multi-terminal net should not be considered equivalent to reducing an edge resulting from inserting a two- terminal net inside a cluster. The average gain that a blockcan take from an n-terminal net i connected to one of its pins, depend- ing on type of the net can be estimated from Table1... ..."

Cited by 6

### Table 1: Routability Gain of a Candidate Block According to a Single Net

"... In PAGE 11: ... Actually, by adding block B to cluster C, an input pin of the cluster gets free and can be used for another net connection. In Table1 this is de ned as in-pin gain. This increases the probability of acceptance of adding the block to the cluster.... In PAGE 11: ... This means that there would be no connection from outside to this pin since all the terminals of the corresponding net are located inside the cluster. Therefore the number of external connections of the cluster, de ned as output congestion gain in Table1 , decreases. This yields less congestion among the clusters.... In PAGE 12: ...As explained above, by considering just the number of shared inputs and outputs as in Equation 5, the packing algorithm cannot di erentiate among the candidate blocks which have di erent impacts on routability. All possible cases yielding di erent total gains are presented in Table1 for one net connected to a candidate block. By incorporating the other routability factors, the gain for each logic block B going into cluster C can be computed as the weighted combination of di erent routability factors as follows: Gain(B; C) = f(Nets(B); Nets(C)) = X i2Nets(B) g(i; Nets(C); B); (6) where g(i; C; B) = 8 gt; gt; gt; gt; gt; gt; lt; gt; gt; gt; gt; gt; gt; : 1 + a fin(P(i; B); P(i; C)) +b fo(P(i; B); P(i; C)) i 2 Nets(C) ?1 c T(i; B) otherwise fin(P(i; B); P(i; C)) is de ned as the gain obtained in input pins of cluster C as de ned in Table 1.... In PAGE 12: ... Therefore, reducing an edge from a multi-terminal net should not be considered equivalent to reducing an edge by inserting a two-terminal net inside a cluster. The average gain that a block can take from an n-terminal net i connected to one of its pins, depending on type of the net can be estimated from Table1 as follows: Gainavg(i; n) = 8 gt; gt; gt; lt; gt; gt; gt; : 2 n = 2 5 3 1 n + 2 3 n?1 n n gt; 2 (7) According to Equation 7, the average gain obtained from a two-terminal net is the highest. This implies that the algorithm gives priority to pushing a two-terminal net entirely inside a cluster as compared to reducing a pin of a multi-terminal net.... In PAGE 15: ... Each path is a chain of output-to-input pin-to-pin connections between a set of blocks (See Figure 5). According to routability gain function ( Table1 ), the output-to-input connection has a high routability gain. When a connection is marked to be critical, it means that there is a long chain of input-output connectivity from this point to the rest of the design.... ..."

### Table 2 lists the citations by subject, using the same keywords as in Table 1. The citations for each subject are chronology ordered.

"... In PAGE 15: ...SOME OBSERVATIONS AND AVENUES FOR RESEARCH 13 Table2 : Subject List of Citations 1977-89 1990-97 Assignment (generalized) [92] [107] Bibliography [52, 120, 124] Branch and bound [89, 90, 95, 62, 88, 93] [98, 102] Complexity [91, 21, 58, 16, 36] [125, 121, 122, 115, 34, 118] Cutting planes [61, 75, 3, 95, 62] Dynamic programming [51, 27, 123, 69] [121] Environment model [65, 113] [25] Facility location [92, 130] [79] Feasibility [96, 54, 55, 60] Fleet mix [67] Heuristic [65, 66] [76, 97] Hydropower model [114] Knapsack [92, 66] [19, 9, 60] Logical [46, 69] [29, 63] Lot-sizing [94] [125, 121] Matching [57, 129, 4, 30] [2] Matroid [84, 33, 31] Multicriteria [123, 135] [64, 25] Multi-terminal ow [57] [87] Nonlinear [91, 90, 27, 6, 102] [23] Price [136, 74] [28, 134] Scheduling [92, 104] [125, 105, 76, 18, 78] [80, 108, 109, 110] Sets [36] [35, 38] Shortest path/route [57, 101, 73, 58] [17] Stability [49, 50, 104] [83, 105, 42, 78, 106, 107] [18, 70, 86, 85, 22, 71, 72] [115, 116, 119, 80, 108] [109, 110] Survey [45, 92, 131, 57] [68, 125, 107, 10, 44] [133, 109, 115, 134] Target analysis [47, 48] TSP [130] [82, 83, 38, 117, 107, 42, 70] [71, 72, 115, 116, 118, 119] Trees [24, 56, 57, 112, 1] [32, 35, 17, 37, 41, 107] [77, 36] [31, 40, 39, 103] Unboundedness [20] Uncertainty [81] [80] Value function [100, 11, 12, 43, 57, 5] [38, 8, 127, 128, 132, 111] [136, 13, 65, 132, 99] [59, 6, 14, 15, 16, 26]... ..."

### Table 1: Routability Gain of a Candidate Block According to a Single Net

"... In PAGE 11: ... Actually,by adding block B to cluster C, an input pin of the cluster gets free and can be used for another net connection. In Table1 thisisde nedasin-pin gain. This increases the probability of acceptance of adding the block to the cluster.... In PAGE 11: ... This means that there would be no connection from outside to this pin since all the terminals of the corresponding net are located inside the cluster. Therefore the number of external connections of the cluster, de ned as output congestion gain in Table1 , decreases. This yields less congestion among the clusters.... In PAGE 12: ...As explained above, by considering just the number of shared inputs and outputs as in Equation 5, the packing algorithm cannot di erentiate among the candidate blocks which have di erent impacts on routability. All possible cases yielding di erent total gains are presented in Table1 for one net connected to a candidate block. By incorporating the other routability factors, the gain for each logic block B going into cluster C can be computed as the weighted combination of di erent routability factors as follows: Gain(B;;C) = f(Nets(B);;Nets(C)) = X i2Nets(B) g(i;; Nets(C);;B);; (6) where g(i;; C;; B)= 8 gt; gt; gt; gt; gt; gt; lt; gt; gt; gt; gt; gt; gt; : 1+a f in (P(i;; B);;P(i;; C)) +b f o (P(i;; B);;P(i;; C)) i 2 Nets(C) ;1 c T(i;; B) otherwise f in (P(i;; B);;P(i;; C)) is de ned as the gain obtained in input pins of cluster C as de ned in Table 1.... In PAGE 12: ... Therefore, reducing an edge from a multi-terminal net should not be considered equivalent to reducing an edge by inserting a two-terminal net inside a cluster. The average gain that a block can takefromann-terminal net i connected to one of its pins, depending on type of the net can be estimated from Table1 as follows: Gain avg (i;; n)= 8 gt; gt; gt; lt; gt; gt; gt; : 2 n =2 5 3 1 n + 2 3 n;1 n n gt;2 (7) According to Equation 7, the average gain obtained from a two-terminal net is the highest. This implies that the algorithm gives priority to pushing a two-terminal net entirely inside a cluster as compared to reducing a pin of amulti-terminal net.... In PAGE 15: ... Each path is a chain of output-to-input pin-to-pin connections between a set of blocks (See Figure 5). According to routability gain function ( Table1 ), the output-to-input connection has a high routability gain. When a connection is marked to be critical, it means that there is a long chain of input-output connectivity from this point to the rest of the design.... ..."

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