Abstract-The traditional way of approaching placement problems in computer-aided design (CAD) tools for analog layout is to explore an extremely large search space of feasible or unfeasible placement configurations, where the cells are moved in the chip plane (being even allowed to overlap in possibly illegal ways) by a stochastic optimizer. This paper presents a novel exploration technique for analog placement operating on a subset of tree representations of the layout-called symmetric-feasible, where the typical presence of an arbitrary number of symmetry groups of devices is directly taken into account during the search of the solution space. The computation times exhibited by this novel approach are significantly better than those of the algorithms using the traditional exploration strategy. This superior efficiency is partly due to the use of segment trees, a data structure introduced by Bentley, mainly used in computational geometry.

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-The traditional way of approaching device-level placement problems for analog layout is to explore a huge search space of absolute placement representations, where cells are allowed to illegally overlap during their moves [3,10]. This paper presents a novel exploration technique for analog placement, operating on the set of tree representations of the layout [6,2], where the typical presence of an arbitrary number of symmetry groups of devices is directly taken into account during the search of the solution space. The efficiency of the novel approach is due to the use of red-black interval trees [4], data structures employed to support operations on dynamic sets of intervals.

-This paper presents a novel algorithm for device-level analog placement with symmetry constraints. Based on the exploration of symmetric-feasible (S-F) binary tree representations [2] of the layout, the novel approach employs 1-3 deterministic skip lists [7,10], exhibiting running times at least 20-30% better than previous (nonslicing) topological algorithms for analog placement, and significantly better (typically, over 100%) than more traditional approaches based on the absolute representation.

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-The traditional way of approaching placement problems in computer-aided design (CAD) tools for analog layout is to explore an extremely large search space of feasible or unfeasible placement configurations, where the cells are moved in the chip plane (being even allowed to overlap) by a stochastic optimizer [2,6]. This paper presents a novel analog placement technique operating on the set of tree representations of the layout [4], where the typical presence of an arbitrary number of symmetry groups of devices is directly taken into account during the exploration of the solution space. The computation times exhibited by this novel approach are typically 3 -6 times better than those of the algorithms using the traditional exploration strategy. This superior efficiency is due to the use of red-black trees, a data structure introduced by Guibas and Sedgewick [3] to support operations on dynamic sets of intervals.

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