Budget Management with Applications

Transcription

1 Algorithmica (2002) 34: DOI: /s Algorithmica 2002 Springer-Verlag New York Inc. Budget Management with Applications Chunhong Chen, 1 Elaheh Bozorgzadeh, 2 Ankur Srivastava, 2 and Majid Sarrafzadeh 2 Abstract. Given a directed acyclic graph with timing constraints, the budget management problem is to assign to each vertex an incremental delay such that the sum of these delays is maximized without violating given constraints. We propose the notion of slack sensitivity and budget gradient to demonstrate the characteristics of budget management. We develop a polynomial-time algorithm for the budget management problem, based on the maximum independent set of an established transitive graph. We show the comparison of our approach with the well-known zero-slack algorithm, and extend it to general weighted graphs. Applications to a class of problems in VLSI CAD are also discussed. Key Words. Budget management, Slack assignment, Directed acyclic graphs, Transitive graphs, Maximum independent set, Timing constraints, VLSI CAD. 1. Introduction. Consider a directed acyclic graph G = (V, E), where V is a set of vertices and E is a set of directed edges. A set of sources (or primary inputs), I V,isa set of vertices without incoming edges, and a set of sinks (or primary outputs), O V, is a set of vertices without outgoing edges. Given I signals each starting from a source I i I at the time a(i i ), we consider their propagation towards O along directed edges and vertices in G, assuming that each vertex v V is associated with a delay d(v) 3 which represents the time it takes for a signal to pass through v and that there is no delay on edges. 4 The latest time of signals to arrive at the output of any vertex v V\I is given recursively by (1) a(v) = max (a(u) + d(v)), u FI(v) where FI(v) is a set of (immediate) predecessors (or fanins)ofv, and a(v) is called the arrival time of v. If the signal at a sink O j O is required to arrive by the time r(o j ), it is required to arrive at the output of any vertex v V\O by the time (2) r(v) = min (r(w) d(w)), w FO(v) where FO(v) is a set of (immediate) successors (or fanouts)ofv, and r(v) is called the required time of v. The slack, s(v), for vertex v is the difference between r(v) and a(v), i.e., s(v) = r(v) a(v). 1 Department of Electrical and Computer Engineering, University of Windsor, Ontario, Canada N9B 3P4. cchen@uwindsor.ca. 2 Computer Science Department, University of California, Los Angeles, CA 90095, USA. {apple, majid}@ cs.ucla.edu. 3 d(v) is assumed to be zero for any v I O. 4 In some applications, edge delay, if any, can be included in vertex delay. Received May 20, 2000; revised November 8, Communicated by C. K. Wong. Online publication July 24, 2002.

2 262 C. Chen, E. Bozorgzadeh, A. Srivastava, and M. Sarrafzadeh The delay distribution of G is a vector of delays D(V) = [d(v 1 ) d(v 2 ) d(v n )] (where n = V ). Similarly, a vector of slacks S(V) = [s(v 1 ) s(v 2 ) s(v n )] is called the slack distribution, and S(V) = n k=1 s(v k) is called the total slack (denoted by S t ). G is said to be (timing) safe if and only if S(V) 0, where 0 is an n-dimensional vector with all elements being zero. Throughout the paper, the graph under consideration is assumed to be safe initially, unless otherwise stated. In fact, if G is initially unsafe by this definition, it can be made safe by, for example, relaxing/increasing the required times for some (or all) sinks. (If this were not possible, one could re-define the safety of a graph as follows: G is said to be safe if and only if S(V) [s min s min s min ], where s min < 0 is the minimum slack in G. This guarantees the initial safety of the graph.) If an incremental delay d(v) > 0 is assigned to any vertex v V, the slack distribution is updated, depending upon the topology of G. In other words, we are obtaining a budget from G by assigning incremental delays to V. This assignment procedure is known as budget management. In general, we define the budget management (or slack assignment) instance (or instance, for brevity) to be a vector of incremental delays D(V) = [ d(v 1 ) d(v 2 ) d(v n )] > 0 which updates the delay distribution from D(V) tod (V) = D(V) + D(V), and hence updates the slack distribution from S(V) tos (V) (the terms budget management and slack assignment will be used interchangeably throughout the paper). In real applications there is a guarantee that incremental delays are non-negative, as long as all vertices delays are initially set to their minimum values. Therefore, negative incremental delays are not considered in this work. Since the slack distribution is strongly related to the arrival times and required times for all vertices, obtaining S (V) is a non-trivial task which is dealt with in the next section. If, after the budget management, G is still safe (i.e., S (V) 0), then the instance is said to be effective and D(V) = n k=1 d(v k) is thereby called the effective budget on G. Let be a set of all possible effective instances for G. We show that (i.e., the number of possible effective instances) can be unlimited. Consider a fragment of a graph with three vertices, as shown in Figure 1, where each vertex has one unit delay. Obviously, there exist numerous effective instances with different effective budgets, a few of which are shown in Figure 1. We are interested in identifying a specific instance (i.e., [5 5 0] in this case) with maximum effective budget. Instead of listing all instances for this Fig. 1. Possible effective budget management instances.

3 Budget Management with Applications 263 Fig. 2. A directed acyclic graph with an optimal budget management instance m D(V). case, we observe that it is generally impractical to find a maximum budget by exhaustive enumeration of instances on G. An optimal budget management instance (denoted by m D(V)) is an effective instance which results in the maximum effective budget B m = m D(V) =max D(V ) D(V ). Intuitively, a maximum effective budget represents the largest amount of incremental delay which can be obtained from G while keeping its safety. This budget can be used for different purposes, depending on applications. While a modified version of this budgeting problem may be required to describe exactly real-world problems, we focus on exploring the optimal budget management itself. Application issues are discussed later in the paper. Figure 2 shows a directed acyclic graph and an optimal budget management instance thereof. The well-known zero-slack algorithm (ZSA) for slack assignment has been proposed [1] with the goal of generating performance constraints for VLSI layout design. Some improved versions of the ZSA can be found in [2] and [3]. In [4] a convex programming problem formulation was given to tackle the delay budgeting problem which is equivalent to slack assignment. More recently, a delay-budgeting approach was presented [5] for timing-closure-driven design. While these prior works focused on applications in VLSI (i.e., very-large-scale integrated circuits) layout, the main disadvantage is that they are not able to provide an optimal solution. In this work we analyze some properties of slack, present an algorithm for budget management, and extend its applications to more general optimization domains. The remainder of the paper is organized as follows. In Section 2 we describe some definitions and characteristics of the budget management problem. Section 3 presents a heuristic algorithm and its extension. Applications of our approach to a class of optimization problems in VLSI Computer-Aided Design (CAD) are given in Section 4. Section 5 concludes the paper.

4 264 C. Chen, E. Bozorgzadeh, A. Srivastava, and M. Sarrafzadeh Fig. 3. Slack distribution and budget management. 2. Analysis of Slack. In this section we discuss some properties of slack and explore the basic paradigm of the budget management problem Basic Characteristics of Budget Management. Consider a safe directed acyclic graph G = (V, E). Without loss of generality, we assume, for convenience of discussion, that the arrival times for all sources are zero and that the required times for all sinks are T spec, a user-specified timing constraint on G. (Note that this assumption is not required in our work. The arrival time and required time for any vertex v V are given by (1) and (2), respectively.) From a practical standpoint, any budget management instance on G is required to satisfy the timing constraint. Thus, only effective budget management instances are of interest. A property of budget management on G is that different instances can lead to the same slack distribution. Figure 3 shows a fragment of an example graph, where the delay distribution is initially assumed to be D(V) = [ ] (note that I and O are not shown in the figure). One can verify that 1 D(V) = [ ] and 2 D(V) = [ ] are two different effective instances on the graph. However, the resulting slack distributions are the same, i.e., S 1 (V) = S 2 (V) = 0. Furthermore, it is observed that for an optimal budget management instance m D(V), the resulting slack distribution must be zero, i.e., S m (V) = 0. This is due to the fact that for any instance, if there exists a slack s(v i )>0 in the resulting slack distribution, one is always able to get a larger effective budget by assigning to vertex v i an incremental delay d i = s(v i ) such that s(v i ) becomes zero. In other words, we have the following lemma (the proof is straightforward): LEMMA 2.1. For a directed acyclic graph G = (V, E), if a budget management instance m D(V) is optimal, then the resulting slack distribution S m (V) = 0. Since s(v i ) is the upper bound of incremental delay for vertex v i while keeping safety of G, the incremental delay which an effective instance can assign to any vertex is no more than its slack. Therefore, any effective budget on G is no more than the total slack of G. This suggests the following lemma. LEMMA 2.2. For a directed acyclic graph G, its total slack, S t, is the upper bound of the maximum effective budget, B m, which results from an optimal budget management instance on G.

5 Budget Management with Applications 265 The maximum effective budget represents the real amount of total incremental delays one can get from G while keeping its safety. If we assume that all incremental delays can be used uniformly for specific optimization purposes, the effective budget would be the exact cost function to be maximized, and an optimal budget management instance stands for the best solution. This assumption, however, may not be true, depending upon applications which are discussed in Section 4. For the cases where the assumption does not hold, we provide an extended version of the budget management problem on weighted graphs (the readers are referred to Section 3). Since calculation of the total slack is straightforward, it can be used as a rough estimate of B m. In general, however, this bound may not be tight, depending upon the specific topologic structure and slack distribution of G (we have more discussions on this later). In Figure 3, for example, the total slack is S t = 20, while the maximum effective budget is B m = Slack Sensitivity. From Figure 3 we see that if an incremental delay of 5 is assigned to v 1 and v 2 each, their slacks are updated to be zero, and so is the slack of their (transitive) fanouts (i.e., v 3 and v 4 ). More generally, for any given graph G = (V, E), if an incremental delay d(v) is assigned to any vertex v V, its slack is decreased by d(v), and its transitive fanins/fanouts may or may not have the reduced slack. Before discussing how d(v) affects the slacks of other vertices, we have the following definitions. DEFINITION 2.1. Assuming that vertex u is an immediate fanin (or fanout) of vertex v, we say that u is slack-sensitive to v (denoted by u v) if an incremental delay (denoted by ε>0) on v results in the reduced slack of u. Weuseu v to represent that u is not slack-sensitive to v. DEFINITION 2.2. A directed path consisting of {v n1,v n2,...,v np } is called a slacksensitive path if v ni and v ni+1 (i = 1, 2,...,p 1) are slack-sensitive to each other (i.e., v ni v ni+1 and v ni+1 v ni ). DEFINITION 2.3. Suppose that vertex u is an immediate fanin of vertex v, we say that u and v are a-controlled if a(v) = a(u) + d(v), orr-controlled if r(v) = r(u) + d(v). Consider two vertices u, v as shown in Figure 4. We look into their slack-sensitivity by considering the following cases: Case A. If u and v are both a-controlled and r-controlled (hence they have the same slack, i.e., s(u) = s(v)), then u v and v u for any ε s(u). Case B. If u and v are a-controlled but not r-controlled (hence s(u) < s(v)), then v u for any ε s(u), while u v for any ε s(v) s(u). Case C. If u and v are r-controlled but not a-controlled (hence s(v) < s(u)), then u v for any ε s(v), while v u for any ε s(u) s(v).

6 266 C. Chen, E. Bozorgzadeh, A. Srivastava, and M. Sarrafzadeh Fig. 4. Slack sensitivity. Case D. If u and v are neither a-controlled nor r-controlled (hence a(v) > a(u) + d(v) and r(u) < r(v) d(v)), then v u for any ε a(v) a(u) d(v) (note that if s(u) s(v), then ε s(u) s(v) implies ε a(v) a(u) d(v)), while u v for any ε r(v) r(u) d(v) (note that if s(v) s(u), then ε s(v) s(u) implies ε r(v) r(u) d(v)). The above slack sensitivity between vertices u and v is summarized in the following lemma: LEMMA 2.3. For graph G = (V, E), u, v V, where u is an immediate fanin of v, their slacks are represented by s(u) and s(v), respectively. (a) When s(u) s(v), one vertex of u and v with larger slack is denoted by u and the other (with less slack) by v (i.e., s(u ) = max{s(u), s(v)}, s(v ) = min{s(u), s(v)}). We have: (a1) v u for ε s(u ) s(v ) (from Cases B, C, and D above). (a2) If u and v are either a-controlled or r-controlled, then u v for ε s(v ) (from Cases B and C above). (a3) If u and v are neither a-controlled nor r-controlled, then u v for ε min{a(v) a(u) d(v), r(v) r(u) d(v)} (from Case D above), where a( ), r( ), and d( ) are the arrival time, required time, and delay, respectively. (b) When s(u) = s(v), we have: (b1) If u and v are a-controlled and hence r-controlled, then u v and v u for ε s(u) (from Case A above). (b2) If u and v are not a-controlled (hence not r-controlled either), then u v and v u for ε min{a(v) a(u) d(v), r(v) r(u) d(v)} (from Case D above). If vertices are slack-sensitive to one another, it follows that during budget management, the total slack reduction (which is 2ε for the two-vertex case as shown in Figure 4) can be much more than the obtained effective budget (which is ε for the two-vertex case). Here, the obtained effective budget can be viewed as a delay benefit, while the reduced slack represents a slack penalty which tends to prevent further delay budgeting. The slack sensitivity facilitates a comprehensive understanding of budget management

7 Budget Management with Applications 267 problems. Intuitively, it is desirable that the budget management instance be able to maximize the benefit and minimize the penalty. This leads us to look at the problem by starting with vertices with maximum slack in the given graph, since vertices with less slack are not slack-sensitive to those with larger slack (see (a1) of Lemma 2.3). For vertices with same slack, we can identify their slack-sensitivity by establishing a so-called slack-sensitive graph where the existence of edge (u,v) implies u v and v u, as described below Transitive Slack-Sensitive Graph. Given a directed graph G = (V, E), we first identify a set of vertices V m V with largest slack s m, i.e., V m ={w w V and s(w) = s m }. Let s m 1 be the second largest slack in G (s m 1 is defined to be zero if all vertices in G have the same slack s m ). We construct a directed slack-sensitive graph G s = (V m, E m ) such that E m ={(u,v) (u,v) E, u v and v u for ε = s m s m 1 }. A transitive slack-sensitive graph G t = (V m, E t m ) of G s is a directed graph such that there is an edge (u,v) E t m if and only if there is a path from u to v in G s. Since G t is always associated with ε, we denote the transitive slack-sensitive graph exactly as G t (ε) = (V m, E t m ), where the assignment of an incremental delay ε to any vertex v V m is reducing the slack of v and all its neighbors u N m (v) ={w (w, v) E t m } by exactly ε. Based on Lemma 2.3, we describe an algorithm for constructing G t (ε) below: G t (ε)-algorithm { /* An algorithm for constructing the transitive slack-sensitive graph */ Input: a directed acyclic graph G = (V, E) Output: a directed transitive graph G t (ε) = (V m, E t m ) Begin Identify V m, a set of vertices with maximum slack s m, and find s m 1, the second largest slack in G; E m,ε s m s m 1 ; /* construct a slack-sensitive graph G s = (V m, E m ) */ for each edge (u,v) E where u,v V m do if u and v are a-controlled or r-controlled then Add an edge (u,v)to E m ; else ε min{ε, a(v) a(u) d(v), r(v) r(u) d(v)}, where a( ), r( ), and d( ) are the arrival time, required time, and delay, respectively; end if end for Construct a transitive slack-sensitive graph G t (ε) = (V m, E t m ) by adding transitive edges to G s = (V m, E m ); End }

8 268 C. Chen, E. Bozorgzadeh, A. Srivastava, and M. Sarrafzadeh According to Lemma 2.3, the selection of ε in the above algorithm guarantees that the slack of any vertex v V m and all its neighbors N m (v) ={w (w, v) E t m } will be reduced by ε if v is assigned an incremental delay ε, and that the slack of any vertex u V\{v, N m (v)} remains unchanged. This is an important consideration in the presentation of our approach for finding an optimal budget management instance in the next section. 3. A Heuristic Algorithm for Budget Management. In this section we present a heuristic algorithm for the budget management problem, which generates a maximum effective budget on any given directed acyclic graph. We also compare it with the conventional ZSA and extend our algorithm to weighted graphs Algorithm Description. Before describing our algorithm, we briefly review the ZSA [1]. The basic idea of the ZSA is to start with vertices with minimum slack and locally perform the slack assignment such that their slacks become zero. This process repeats until the slacks of all vertices become zero. More specifically, at each iteration, the ZSA identifies a path on which all vertices have minimum slack (denoted by s min ), then assigns to each vertex an incremental delay s min /N min, where N min is the number of vertices on the path. In Figure 3, for example, the path (v 1, v 3, v 4 ) is found first, and each of the three vertices is assigned an incremental delay of 5/3 units. The slacks of all vertices except v 2 in the figure become zero, while s(v 2 ) is updated as 5/3. After assigning an incremental delay of 5/3 tov 2 in the second iteration, the slack distribution is updated to be zero, and the algorithm terminates with the effective budget of 20/3 (note that the maximum effective budget of Figure 3 is 10). This example shows that while the ZSA is simple to use and easy to implement, the result is far from the optimum. In contrast, our algorithm begins with a set of vertices, V m, with maximum slack. A transitive slack-sensitive graph G t (ε) = (V m, E t m ) is then constructed. Based on discussions in the above section, a maximum independent set 5 (MIS) [6] of G t (ε) corresponds to a set of vertices V MIS V m such that the number of vertices which are slack-sensitive to any vertex v V MIS is minimized. It should be pointed out that while the MIS problem is NP-hard for general graphs, it is polynomial-time solvable for transitive graphs as discussed in this paper. We assign an incremental delay ε to each vertex in V MIS, and update the slack of vertices in V m. This process of constructing G t (ε), finding MIS, and updating slack continues until the slack distribution becomes zero. Clearly, each step of the process contributes ε V MIS to the final budget. Finding an MIS of G t (ε) is straightforward as it can be done by iteratively identifying a vertex v min with minimum degree in G t (ε). The pseudo-code of our MIS-based algorithm for the budget management problem is 5 A maximum independent set of a graph is a set of independent vertices (i.e., no two vertices are connected by an edge) such that the number of vertices in this set is maximum.

9 Budget Management with Applications 269 given below: Maximum-Independent-Set-based Algorithm (MISA) { /* An algorithm for finding a budget management instance */ Input: graph G = (V, E) and timing constraint T spec Output: a budget management instance D(V) Begin Compute slack for each vertex v V, and find maximum slack s m ; Initialize D(V) 0; While (s m > 0) { Construct a transitive slack-sensitive graph G t (ε) using the G t (ε)- algorithm; Find a maximum independent set, V MIS,ofG t (ε); Assign an incremental delay ε to each vertex in V MIS, i.e., d(w) d(w) + ε, w V MIS ; Update slack distribution and s m ; } End } In the above algorithm, the budget management problem is partitioned into several subproblems which are optimally solved individually. However, there is no guarantee in general that the algorithm is optimal for the original problem since the effects among different algorithmic steps are not accounted for. This heuristic becomes optimal when there are no or minimum effects between the steps. Theorems 3.1 and 3.2 below show two examples of such cases: THEOREM 3.1. For any directed acyclic graph G where all vertices have the same slack and the value of ε is greater than or equals the slack, the MISA produces an optimal budget management instance which results in the maximum effective budget of G. PROOF. Suppose all vertices in G have slack s, and ε = s. We can obtain a transitive slack-sensitive graph G t (s) by the G t (ε)-algorithm (see Section 2.3). All these vertices can be expressed as the union of independent subsets of vertices (where the independent means that all vertices in a subset are not included in any other subsets) such that each of such subsets is a maximal clique which contributes to the final budget by s at most. In particular, since ε = s, if there are several connected components (subgraphs) in G t (s), the slack assignment in one component would be independent of that in another. Let N c be the number of all cliques in G t (s), the maximum effective budget is s N c. Since the cardinality of a maximum independent set in the graph is exactly N c, the MISA provides the effective budget of s N c and hence results in an optimal budget management instance.

10 270 C. Chen, E. Bozorgzadeh, A. Srivastava, and M. Sarrafzadeh THEOREM 3.2. If a given directed acyclic graph is a tree, the MISA produces an optimal budget management instance by simply selecting all leaves of subtrees in the slacksensitive graph (i.e., G t (ε)) as a maximum independent set. Based on the above discussions, we have the following theorem (see [6] and [7] for proof): THEOREM 3.3. The time complexity of the MISA algorithm is O(K n 3 ), where n and K are the number of vertices and the number of distinct slacks in a given graph, respectively. The space complexity of the algorithm is O(n 2 ) in the worst case. In the above theorem, the value of K strongly depends upon the given graphs and their slack distributions. However, our experience shows K n for most applications [8] Dynamic Characteristics of Budget Management. Since slack distribution is related to given timing constraint T spec, the total slack and maximum effective budget of a given graph G = (V, E) are associated with T spec. Let B m (T ) be the maximum effective budget of G with the timing constraint T. We define the budget gradient of the graph: B g = B m (T )/ T which stands for the derivative of B m (T ) with respect to T. Unlike B m which is the static characterization of budget management problems, B g is a measure of the dynamic characteristic of budget management. No matter what timing constraint is specified, we can use the MISA algorithm for G and, at the last iteration of the algorithm, obtain a specific graph G 0 where all vertices have the same slack. G and G 0 have the same topology, but their slack distributions are different. If we use G 0 t to represent the transitive slack-sensitive graph of G 0, then the difference between B m (T a ) and B m (T b ) for G is given by (T a T b ) VMIS 0, where T a and T b are two different timing constraints (T a > T b ), and VMIS 0 is an MIS on G0 t. In other words, we have proved: THEOREM 3.4. For any directed acyclic graph G, its budget gradient is equal to the cardinality of the MIS on the transitive slack-sensitive graph of a specific graph G 0, where G 0 is obtained from G in the last iteration of the MISA algorithm. To illustrate, we calculate, by inspection, the budget gradients for graphs of Figures 2 and 3, which are 3 and 2, respectively. This implies that more budget is achievable from Figure 2 than from Figure 3 by relaxing the timing constraint. In other words, Figure 2 shows a preferable dynamic characteristic during budget management. Recall that the maximum effective budgets for Figures 2 and 3 are 5 and 10, respectively, meaning that Figure 3 has the better static performance in terms of budget management Comparison with the ZSA. To look at the difference between the MISA and ZSA in terms of effective budget, we consider a vertex v with f fanouts as shown in Figure 5, where all vertices are assumed to have the same slack s. The effective budget provided by the ZSA is ZSA D(V) =s + ( f 1)s/2. In contrast, the MISA leads to the maximum effective budget B m = MISA D(V) =fs. Thus, we have the ratio ( MISA D(V) ZSA D(V) )/ ZSA D(V) =( f 1)/( f + 1), which means that MISA D(V) can be as high as almost twice ZSA D(V) when f 1. This shows that the MISA can provide

11 Budget Management with Applications 271 Fig. 5. A vertex with f fanouts each with the same slack s. a significant improvement over the ZSA when there exist a large number of fanouts. In this figure the total slack is S t = ( f + 1) s, and the ratio of B m to S t is f/( f + 1), indicating that S t is a loose upper bound of B m for a small value of f. For instance, B m only accounts for a half of S t when f = 1. Figure 6 shows an eight-vertex example graph, where the initial delay distribution is assumed to be a unit vector, i.e., D(V) = 1. By inspection, the ZSA generates the budget management instance ZSA D(V) = [ ] (refer to Section 3.1 for more details), leading to the effective budget ZSA D(V) =12. By applying the MISA, we first find s m = 5 and ε = 2 in the first iteration. G t (ε) consists of v 1,v 6, and v 7, and its MIS is V MIS ={v 1,v 6,v 7 }. Thus, d(v 1 ) = d(v 6 ) = d(v 7 ) = 2. In the second iteration we have s m = 3 and ε = 3. Since all vertices except v 8 have same slack s m, G t (ε) consists of v 1,v 2,...,v 7, and V MIS ={v 3,v 5,v 6,v 7 }. After assigning an incremental delay, ε, to each vertex in V MIS, we obtain the effective budget management instance MISA D(V) = [ ] with the final slack distribution S MISA (V) = 0. As a result, the maximum effective budget produced by the MISA is B m = MISA D(V) = 18, which is a 50% improvement compared with ZSA D(V) =12. Note that the total slack of this graph is Extension to Weighted Graphs. In the above discussion we choose the effective budget as an objective function to be maximized with the implicit assumption that Fig. 6. Calculation of effective budget.

12 272 C. Chen, E. Bozorgzadeh, A. Srivastava, and M. Sarrafzadeh the budget on each vertex is of the same importance. In the real world, however, this assumption may not be practical. For example, in some applications, vertices in the graph may have an upper bound of delay, and it would not make sense if they were assigned incremental delays which are larger than the bound. To tackle this issue, we propose a modified version of the budget management problem on weighted graphs. In such a graph, each vertex is associated with a positive weight which measures the importance of delay budgeting from it. Thus, the MISA discussed before can be modified by extending the MIS to the maximum weighted independent set 6 (MWIS) on the transitive (weighted) slack-sensitive graph G t (ε). The objective here is to find an optimal weighted budget management instance which leads to the maximum weighted effective budget on the graph. In Figure 1, for example, if it is weighted and the three vertices are assumed to have the following weights, weight(v 1 ) = weight(v 2 ) = 2 and weight(v 3 ) = 5, then the MWIS turns out to be {v 3 }, and the optimal weighted instance is [0 0 5] with a maximum weighted effective budget of 25. It should be pointed out that all lemmas and theorems about unweighted graphs can be extended accordingly to deal with weighted graphs. It has been shown [9] that the MWIS problems on transitive graphs can be solved by the minimum-flow algorithm in O(ne log(n 2 /e)) time, where n and e are the number of vertices and the number of edges, respectively, in the graph. This implies that if weighted graphs describe real-world problems correctly, our method is still optimal. For computational efficiency, one may resort to a heuristic. A faster heuristic for finding MWIS is similar to that of finding MIS, except that instead of starting with a vertex of minimum degree for the case of MIS, we iteratively select a vertex with maximum ratio r = g i /deg i in G t (ε), where g i and deg i are the weight and degree of vertex v i, respectively. Alternatively, one can use r = g i + λ/deg i (where λ is a constant), and try different values of λ towards a high probability of obtaining a reasonably good solution, depending upon the specific graphs under consideration. The time complexity of this heuristic is O(n log n) [6]. 4. Applications. In this section we discuss applications of budget management to a class of constrained optimization problems which arise in design and synthesis of integrated circuits (IC) Gate Sizing. A gate-level implementation of IC can be mapped into a directed acyclic graph G where vertices correspond to logic gates, and edges correspond to interconnects between the gates. The intrinsic delay of gates in the implementation is transformed into vertex delay in G (the delay of interconnects driven by a gate, if accounted for, can be added into the vertex delay). In IC design, circuit area (physical size) has been a primary concern. Given specific timing constraints which represent the circuit performance requirement, a gate sizing problem [10], [11] is to find a set of vertices/gates such that their physical sizes can be reduced (by using smaller cell instances from a target library) for area/power minimization without violating timing constraints. Since 6 A maximum weighted independent set of a (vertex-weighted) graph is a set of independent vertices such that the sum of their weights is maximum.

13 Budget Management with Applications 273 Fig. 7. Area/power reduction by gate sizing. a gate with smaller size has longer delay and hence requires more budget (i.e., more incremental delay), area/power optimization by gate sizing is strongly determined by the budget management instance of G. An optimal budget management instance can lead to significant (total) area/power reduction (note that area/power reduction may not be maximized due to the discrete nature of gate sizes available in the library). Considering the fact that different gates may show different delay area (or delay power) tradeoff curves, it is preferable to define a weighted graph based on these curves, and find an optimal weighted budget management instance on it. Figure 7 illustrates an example circuit, where each gate delay is assumed to be 1. The two gates (shaded in the figure) with high budget could be downsized for area/power reduction while keeping the timing performance of the circuit satisfied Layout Design. Traditionally, budget management is used in the generation of performance constraints for timing-driven placement [1] [3], [12] since the slack of a vertex (gate) imposes an upper bound of delay on the signal nets it drives. On the whole, the maximum effective budget tends to maximize the flexibility/freedom for all signal nets during layout while keeping the timing constraints. This flexibility is very desirable for most placement and routing tools. Whenever necessary, one can solve the budget management problem on a weighted graph to avoid over-estimated timing constraints on particular signal nets, whose driving gates are supposed to be assigned the higher weights. Figure 8 shows a possible layout (placement and routing) of Figure 7. The two gates (shaded in the figure) with higher budget can be placed in many optional locations (e.g., around the right lower corner in the figure) without or with little concern about timing degradation of the circuit. Budget management facilitates the routability of more nets within the timing constraints Wire Sizing. Wire sizing is a popular interconnect-driven optimization technique for high performance [13]. If we use vertices to represent wires in layout, and vertex delay to represent the wire width, the routing topology can be converted into a DAG, where

14 274 C. Chen, E. Bozorgzadeh, A. Srivastava, and M. Sarrafzadeh Fig. 8. Flexibility of signal nets during layout. the delay budget denotes how much the wire width can be increased. Figure 9 shows an example of wire sizing problem in a routing channel where four tracks are available. In the corresponding DAG (see Figure 9(b)), the existence of a directed edge from node v i to v j implies that the corresponding wires (i.e., w i and w j ) are physically adjacent and overlap vertically. Since a wider wire helps improve the timing performance, the best strategy for wire sizing can be obtained by finding an optimal budget management instance on the graph. In this figure the two wires (i.e., w 2 and w 3 ) could be selected to be upsized (note that the wire w 4 should not be selected for upsizing). 5. Conclusion and Future Work. In this paper we have dealt with the budget management problems for directed acyclic graphs by analyzing the slack sensitivity, constructing the transitive slack-sensitive graph, and presenting a heuristic budget management Fig. 9. Wire sizing in a routing channel using budget management.

15 Budget Management with Applications 275 algorithm. The proposed approach is based on an MIS which is polynomial-time solvable for transitive graphs. Several applications of budget management in the VLSI CAD area have been demonstrated. Since the optimal solution of budget management problems requires polynomial time which is still computationally expensive for large-size graphs, fast estimation of maximum effective slack (and/or its tight upper bound) would be very desirable. Further research work includes looking for optimal solutions in general cases. Also we expect to find more applications of budget management on other areas. Acknowledgment. The authors thank the anonymous reviewers for their constructive comments. In particular, they thank one of the reviewers for his/her invaluable feedback and advice on the proof of Theorem 3.1. References [1] R. Nair, C. L. Berman, P. S. Hauge, and E. J. Yoffa, Generation of Performance Constraints for Layout, IEEE Transactions on Computer-Aided Design, CAD-8(8): , August [2] T. Gao, P. M. Vaidya, and C. L. Liu, A New Performance Driven Placement Algorithm, in Proceedings of the International Conference on Computer-Aided Design, pp , [3] H. Youssef and E. Shragowitz, Timing Constraints for Correct Performance, in Proceedings of the International Conference on Computer-Aided Design, pp , [4] G. E. Tellez, D. A. Knol, and M. Sarrafzadeh, A Graph-Based Delay Budgeting Algorithm for Large Scale Timing-Driven Placement Problems, in Proceedings of the Fifth ACM/SIGDA Physical Design Workshop, pp , April [5] C. Kuo and A. C.-H Wu, Delay Budgeting for a Timing-Closure-Design Method, in Proceedings of the International Conference on Computer-Aided Design, pp , [6] R. H. Moehring, Algorithmic Aspects of Comparability Graphs and Interval Graphs, in Graphs and Orders, edited by I. Rival, Reidel, New York and London, pp , May [7] C. Chen and M. Sarrafzadeh, An Effective Algorithm for Gate-Level Power-Delay Tradeoff Using Two Voltages, in Proceedings of the International Conference on Computer Design, pp , [8] C. Chen, A Srivastava, and M. Sarrafzadeh, On Gate Level Power Optimization Using Dual Supply Voltages, IEEE Transactions on VLSI Systems, 9(5): , October [9] D. Kagaris and S. Tragoudas, Maximum Independent Sets on Transitive Graphs and Their Applications in Testing and CAD, in Proceedings of the International Conference on Computer-Aided Design, pp , [10] P. Girard, C. Landrault, S. Pravossoudovitch, and D. Severac, A Gate Resizing Technique for High Reduction in Power Consumption, in Proceedings of the International Symposium on Low Power Electronics and Design, pp , [11] H. R. Lin and T. Hwang, Power Reduction by Gate Sizing with Path-Oriented Slack Calculation, in Proceedings of IEEE ASP-DAC 95/CHDL 95/VLSI 95, pp. 7 12, [12] W. Swartz and C. Sechen, Timing Driven Placement for Large Standard Cell Circuits, in Proceedings of the IEEE/ACM Design Automation Conference, pp , [13] C. Chu and D. F. Wong, Closed Form Solution to Simultaneous Buffer Insertion/Sizing and Wire Sizing, in Proceedings of the International Symposium on Physical Design, pp , 1997.