Approximation algorithms for the bandwidth minimization problem for a large class of trees

We present approximation algorithms for the bandwidth minimization problem (BMP) for a large class of trees. The BMP is NP-hard, even for trees of maximum node degree 3. The problem finds applications in many areas, including VLSI layout, multiprocessor scheduling, and matrix processing, and has been studied for both graphs and matrices. We study the problem on trees having the following property: given any tree nodev, the depth difference of any two nonempty subtrees rooted atv is bounded by a constantk. We call such treesh(k)trees orgeneralized height-balanced (GHB)trees. The above definition extends the class of balanced trees to trees with depthd=Θ(\N\). For any tree in the above defined class, anO (logd) times optimal algorithm is presented. Furthermore, we extend the application of the algorithm to trees that simulate theh(k) property, which we callh(k)-like trees, and also provide intuitive ideas for an approximation algorithm for general trees.     

When a Tritrophic Interaction Goes Wrong to the Third Level: Xanthoxylin From Trees Causes the Honeybee Larval Mortality in Colonies Affected by the River Disease

The “River Disease” (RD), a disorder impacting honeybee colonies located close to waterways with abundant riparian vegetation (including Sebastiania schottiana, Euphorbiaceae), kills newly hatched larvae. Forager bees from RD-affected colonies collect honeydew excretions from Epormenis cestri (Hemiptera: Flatidae), a planthopper feeding on trees of S. schottiana. First-instar honeybee larvae fed with this honeydew died. Thus, we postulated that the nectars of RD-affected colonies had a natural toxin coming from either E. cestri or S. schottiana. An untargeted metabolomics characterization of fresh nectars extracts from colonies with and without RD allowed to pinpoint xanthoxylin as one of the chemicals present in higher amounts in nectar from RD-affected colonies than in nectars from healthy colonies. Besides, xanthoxylin was also found in the aerial parts of S. schottiana and the honeydew excreted by E. cestri feeding on this tree. A larva feeding assay where xanthoxylin-enriched diets were offered to 1^st instar larvae showed that larvae died in the same proportion as larvae did when offered enriched diets with nectars from RD-colonies. These findings demonstrate that a xenobiotic can mimic the RD syndrome in honeybee larvae and provide evidence of an interspecific flow of xanthoxylin among three trophic levels. Further, our results give information that can be considered when implementing measures to control this honeybee disease.

     

Resource requirements and layouts for field programmableinterconnection chips

Field-programmable interconnection chips (FPIC's) provide the capability of realizing user programmable interconnection for any desired permutation. Such an interconnection is very much desired for supporting rapid prototyping of hardware systems and for providing programmable communication networks for parallel and distributed computing. An FPIC should realize any possible permutation of input to output pins via a set of programmable switches. In this paper, we show that any such architecture requires a minimum of Ω(n log n) switches, where Ω is the number of I/O pins. The result stems from an analysis of the underlying permutation network. In addition, for networks of bounded degree d, we prove an Ω(log_d-1 n) bound on the routing delay (maximum length of routing paths for specific I/O permutations) and an Ω(n log_d-1 n) bound on the average utilization of programmable switches used by the FPIC to implement a specific permutation. For the same type of networks, we prove an Ω(n log_d-1 n) bound on the number of nodes of the network. Furthermore, we design efficient architectures for FPIC's offering a wide variety of routing delays, high average programmable resource utilization, and O(n²)-area two-layer layouts. The proposed structures are called hybrid Benes-Crossbar (HBC) architectures and clearly exhibit a tradeoff between performance (routing delay utilization) and area of the layout     

Bandwidth Minimization: An approximation algorithm for caterpillars

The Bandwidth Minimization Problem (BMP) is the problem, given a graphG and an integerk, to map the vertices ofG to distinct positive integers, so that no edge ofG has its endpoints mapped to integers that differ by more thank. There is no known approximation algorithm for this problem, even for the case of trees. We present an approximation algorithm for the BMP for the case of special graphs, called caterpillars. The BMP arises from many different engineering applications which try to achieve efficient storage and processing and has been studied extensively, especially with relation to other graph layout problems. In particular, the BMP for caterpillars is related to multiprocessor scheduling. It has been shown to be NP-complete, even for degree-3 trees. Our algorithm, gives a logn times optimal algorithm, wheren is the number of nodes of the caterpillar. It is based on the idea of level algorithms.This work has been, in part, supported by the Computer Learning Research (CLEAR) Center at UTD.     

Approximation algorithms for the bandwidth minimization problem for a large class of trees

We present approximation algorithms for the bandwidth minimization problem (BMP) for a large class of trees. The BMP is NP-hard, even for trees of maximum node degree 3. The problem finds applications in many areas, including VLSI layout, multiprocessor scheduling, and matrix processing, and has been studied for both graphs and matrices. We study the problem on trees having the following property: given any tree nodev, the depth difference of any two nonempty subtrees rooted atv is bounded by a constantk. We call such treesh(k)trees orgeneralized height-balanced (GHB)trees. The above definition extends the class of balanced trees to trees with depthd=Θ(\N\). For any tree in the above defined class, anO (logd) times optimal algorithm is presented. Furthermore, we extend the application of the algorithm to trees that simulate theh(k) property, which we callh(k)-like trees, and also provide intuitive ideas for an approximation algorithm for general trees. This work has been supported in part by the Computer Learning Research (CLEAR) Center at the University of Texas at Dallas.