1. Windowing 290N: The Unknown Component Problem Lecture 10

2. Outline  Definitions  Window computation  Generalization to sequential networks

3. Windowing  A window includes k levels of TFI k levels of TFI m levels of TFO m levels of TFO all re-convergent paths captured in this scope all re-convergent paths captured in this scope Window POs Window PIs k = 3 m = 3

4. Definition of Leaves and Roots  Definition. Given a directed acyclic graph and two non-overlapping subsets of its nodes, one set is called the leaf set and the other the root set, if every path from any node in the root set towards the sources of the graph passes through some node in the leaf set.

5. Definition of a Window  Definition. Given a directed acyclic graph and two subsets of its nodes, which are in the leaves/roots relationship, a window is a subset of nodes of the graph, which contains the roots and all nodes between the root set and the leaf set. The nodes in the leaf set are delimiters and do not belong to the set of nodes included in the window.

6. Definition of Distance  Definition. A path between a pair of nodes is distance-k if it spans exactly k edges between the pair.  Definition. Two nodes are distance-k from each other if there exists a distance-k path between them, and there exists a distance-k path between them, and all other paths between them are distance-k or less. all other paths between them are distance-k or less.

7. Window Computation nodeset Window( node N, int nFanins, int nFanouts ) { nodeset I1 = CollectNodesTFI( {N}, nFanins ); nodeset I1 = CollectNodesTFI( {N}, nFanins ); nodeset O1 = CollectNodesTFO( {N}, nFanouts ); nodeset O1 = CollectNodesTFO( {N}, nFanouts ); nodeset I2 = CollectNodesTFI( O1, nFanins + nFanouts ); nodeset I2 = CollectNodesTFI( O1, nFanins + nFanouts ); nodeset O2 = CollectNodesTFO( I1, nFanins + nFanouts); nodeset O2 = CollectNodesTFO( I1, nFanins + nFanouts); nodeset S = I2  O2; nodeset S = I2  O2; nodeset L = CollectLeaves( S ); nodeset L = CollectLeaves( S ); nodeset R = CollectRoots( S ); nodeset R = CollectRoots( S ); return ( L, R ); return ( L, R );}

8. Example O1O1 O1O1 I1I1 I1I1 S N S S S