Mojom frontend: An algorithm for detecting ill-founded nodes in a graph.

mojom/mojom_parser/utils/wellfounded_graphs.go

+// Copyright 2016 The Chromium Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style license that can be
+// found in the LICENSE file.
+
+package utils
+
+import (
+        "bytes"
+        "fmt"
+)
+
+// This file contains an implementation of an algorithm to check the well-foundedness
+// of two-sorted directed graphs. See the paper
+// "Well-Founded Two-Sorted Directed Graphs" (https://goo.gl/ipxFKu) for background
+// and detailed definitions. Here we give only a high-level overview.
+//
+// A two-sorted graph is a directed graph that contains two sorts of nodes called circle nodes and
+// square nodes. A node in a two-sorted graph is *well-founded* iff it satisfies the
+// following recursive definition:
+// (i) Leaf nodes are well-founded
+// (ii) A circle node is well-founded iff all of its children are well-founded
+// (iii) A non-leaf square node is well-founded iff at least one of its children is well-founded.
+// (See the paper for a more logically correct definition.)
+//
+// See the comments on the main function |CheckWellFounded| below for a description
+// of the algorithm.
+
+///////////////////////////////////////////////////
+// Node type
+//
+// A |Node| is a node in a directed graph.
+///////////////////////////////////////////////////
+
+type Node interface {
+        // Returns the list of children of this node.
+        OutEdges() []Node
+
+        // Returns whether or not this node is a square node.
+        IsSquare() bool
+
+        // SetKnownWellFounded is invoked by the algorithm in order to set the
+        // fact that the algorithm has determined that this node is well-founded.
+        // An implementation of |Node| should cache this--even between different
+        // invocations of the algorithm on different starting nodes--and return
+        // |true| from the method |KnownWellFounded| just in case this method has
+        // ever been invoked.  In this way any nodes verified to be well-founded during
+        // one run of the algorithm do not need to be checked during a later run of the
+        // algorithm on a different starting node.
+        SetKnownWellFounded()
+
+        // Returns whether or not the function |SetKnownWellFounded| has ever
+        // been invoked on this node, during the lifetime of a program using
+        // this library.
+        KnownWellFounded() bool
+
+        // Returns the name of this node, appropriate for debugging.
+        DebugName() string
+}
+
+///////////////////////////////////////////////////
+// CheckWellFounded function
+//
+// This is the main public function.
+///////////////////////////////////////////////////
+
+// CheckWellFounded checks whether the sub-graph rooted at |root| contains any ill-founded nodes.
+// If any ill-founded nodes are detected then a non-nil |CycleDescription| is returned containing
+// a cycle of ill-founded nodes. If every node is well-founded then nil is returned.
+//
+// If the graph contains only circle nodes then well-foundedness is equivalent to
+// acyclicality and so the returned cycle is a proof that the nodes contained in it
+// are ill-founded. But in general (if the graph contains square nodes) the returned
+// |CycleDescription| is only meant to serve as an example of some of the
+// ill-foundedness contained in the subgraph. The cycle is guaranteed to contain only
+// ill-founded nodes and the cycle may be considered part of a proof that thes nodes
+// are in fact ill-founded. But the cycle does not necessarily contain every
+// ill-founded node and it does not
+// necessarily constitute a complete proof of the ill-foundedness
+// because a graph that contains square nodes is allowed to contain some cycles and still be
+// well-founded. The intended application of the returned CycleDescription is to be used to
+// describe to a user the location of the ill-foundedness in order to allow the user to modify
+// the graph in order to make it well-founded.
+//
+// This function may be invoked multiple times on different starting nodes during the life
+// of a program. If the function is invoked once on node |x| and no ill-foundedness is
+// found and then the function is invoked later on node |y| and if node |z| is reachable
+// from both |x| and |y| then node |z| will be marked |KnownWellFounded| during the
+// first run of the function and so the graph below |z| will not have to be inspected
+// during the second run of the function.
+//
+// Our algorithm proceeds in three phases:
+// (1) Phase 1 consists of a depth-first traversal of the graph whose purpose is two-fold:
+//     (a) To prove directly that as many nodes as possible are well-founded and mark them
+//         so by invoking SetKnownWellFounded().
+//     (b) To prepare for phase 2 by constructing two sets of nodes called the |foundationSet| and
+//         the |pendingSet|
+//     See the comments at |checkWellFoundedPhase1| for more details.
+//
+// In many cases phase 1 will be able to prove that every node is well-founded and so the algorithm
+// will terminate without entering phase 2. This is the case for example if the graph has no
+// cycles at all.
+//
+// (2) The purpose of phase 2 is to propogate the |KnownWellFounded| property to the remaining well-founded
+//     nodes (the ones that could not be verified as well-founded during phase 1.) Phase 2 proceeds in
+//     multiple rounds. During each round the |KnownWellFounded| property is propogated from the
+//     |foundationSet| to the |pendingSet|. See the comments at |checkWellFoundedPhase2| for more details.
+//
+//  If there are no ill-founded nodes then the algorithm terminates after phase 2.
+//
+// (3) Phase 3 of the algorithm consists of building a |CycleDescription| in the case that there are ill-founded
+//     nodes.  See the method |findKnownCycle|.
+
+func CheckWellFounded(root Node) *CycleDescription {
+        return checkWellFounded(root, nil)
+}
+
+// checkWellFounded is a package-private version of CheckWellFounded intendend to be invoked by tests.
+// It offers a second parameter |debugDataRequest|. If this is non-nil then its fields will be filled
+// in with debugging data about the output of phase 1.
+func checkWellFounded(root Node, debugDataRequest *debugData) *CycleDescription {
+        if root == nil {
+                return nil
+        }
+        finder := makeCycleFinder()
+        finder.debugDataRequest = debugDataRequest
+        holder := finder.holderForNode(root)
+        return finder.checkWellFounded(holder)
+}
+
+///////////////////////////////////////////////////
+/// cycleFinder type
+///
+/// A cycleFinder is an object that holds state during the execution of the algorithm.
+///////////////////////////////////////////////////
+type cycleFinder struct {
+
+        // Maps each node seen by the computation to its holder.
+        nodeToHolder map[Node]*nodeHolder
+
+        foundationSet, pendingSet NodeSet
+
+        visitationIndex int
+        currentPath     nodeStack
+
+        debugDataRequest *debugData
+}
+
+type debugData struct {
+        initialPendingSet    []Node
+        initialFoundationSet []Node
+}
+
+func makeCycleFinder() cycleFinder {
+        finder := cycleFinder{}
+        finder.nodeToHolder = make(map[Node]*nodeHolder)
+        finder.foundationSet = MakeNodeSet()
+        finder.pendingSet = MakeNodeSet()
+        finder.currentPath = nodeStack{}
+        return finder
+}
+
+// checkWellFounded contains the top-level implementation of the algorithm.
+func (finder *cycleFinder) checkWellFounded(nodeHolder *nodeHolder) *CycleDescription {
+        if nodeHolder.node.KnownWellFounded() {
+                // This node is already known to be well-founded because of an earlier
+                // execution of the algorithm.
+                return nil
+        }
+
+        // Perform phase 1.
+        finder.checkWellFoundedPhase1(nodeHolder)
+
+        // In tests we pass back some debugging information here.
+        if finder.debugDataRequest != nil {
+                finder.debugDataRequest.initialPendingSet = finder.pendingSet.ToNodeSlice()
+                finder.debugDataRequest.initialFoundationSet = finder.foundationSet.ToNodeSlice()
+        }
+
+        // All nodes have been verified as well-founded.
+        if finder.pendingSet.Empty() {
+                return nil
+        }
+
+        // Perform phase 2.
+        finder.checkWellFoundedPhase2(nodeHolder)
+
+        // All nodes have been verified as well-founded.
+        if finder.pendingSet.Empty() {
+                return nil
+        }
+
+        // If we are here then there is at least one ill-founded node.
+
+        // In order to build a canonical cycle description, find the illfounded
+        // node with the least visitation order.
+        var minVisitationOrder int
+        var minIllfoundedNode Node = nil
+        for n, _ := range finder.pendingSet.elements {
+                if minIllfoundedNode == nil || n.visitationOrder < minVisitationOrder {
+                        minVisitationOrder = n.visitationOrder
+                        minIllfoundedNode = n.node
+                }
+        }
+
+        // Starting from the ill-founded node with the least visitation order,
+        // build a canonical cycle.
+        freshCycleFinder := makeCycleFinder()
+        holder := freshCycleFinder.holderForNode(minIllfoundedNode)
+        return freshCycleFinder.findKnownCycle(holder)
+}
+
+// checkWellFoundedPhase1 is a recursive helper function that does a depth-first traversal
+// of the graph rooted at |nodeHolder|. The goal of phase 1 is to mark as many
+// of the nodes as possible as |KnownWellFounded| and to set up the |pendingSet|,
+// the |foundationSet|, and the |parentsToBeNotified| sets so that the remaining
+// well-founded nodes will be marked as |KnownWellFounded| in phase 2.
+//
+// In more detail the following steps are performed for each node x:
+// (a1) If it can be verified during the traversal that x is well-founded then
+// x will be marked as |KnownWellFounded|. This occurs if x is a leaf, or x
+// is a circle and each child node of x is known well-founded before being
+// visited as a child of x, or x is a square and at-least one child node of x
+// is known well-founded before being visited as a child of x.
+// (a2) Otherwise if it cannot be determined during traveral that x is well-founded then
+//   (i) x is added to the |pendingSet|.
+//   (ii) x is added to the |parentsToBeNotified| set of all of its children.
+// (b) In step (a1) if at the time x is found to be well-founded x already has
+// some parent node x' in its |parentsToBeNotified| set (meaning that step a2 occurred
+// earlier for x' and so x' is in the |pendingSet|) then x is added to the |foundationSet|.
+// In phase 2, the fact that x is in the foundation set and x' is in the pending set will be
+// used to propogate known-wellfoundedness to x'.
+func (finder *cycleFinder) checkWellFoundedPhase1(nodeHolder *nodeHolder) {
+        if nodeHolder.node.KnownWellFounded() {
+                // This node is known to be well-founded before being visited.
+                // This occurs when the node was marked |KnownWellFounded| during a
+                // previous run of the algorithm. It follows that all nodes reachable
+                // from this node have also been so marked. We therefore don't need
+                // to traverse the part of the graph below this node during this run
+                // of the algorithm and so we treat this node as a leaf node.
+                nodeHolder.state = vsVisitEnded
+                return
+        }
+
+        // Mark the visit as started.
+        nodeHolder.state = vsVisitStarted
+
+        // Next we examine each of the children and recurse into the unvisited ones.
+        sawUnverifiedChild := false
+        for _, child := range nodeHolder.node.OutEdges() {
+                childHolder := finder.holderForNode(child)
+                if childHolder.state == vsUnvisited {
+                        // Recursively visit this child.
+                        finder.checkWellFoundedPhase1(childHolder)
+                }
+
+                // After having visited a child we use the results to update the status of this node.
+                // We could express the logic here more concisely, but  the logic is easier
+                // to understand if we treat circles and squares seperately,
+                if nodeHolder.node.IsSquare() {
+                        if nodeHolder.node.KnownWellFounded() {
+                                // This square node has already been marked |KnownWellFounded| becuase
+                                // of an earlier iteration through this loop. There is nothing else to do.
+                                continue
+                        }
+                        if childHolder.node.KnownWellFounded() {
+                                // We mark a square node as |KnownWellFounded| as soon as we can so
+                                // that if any of its descendants are also parents, the well-foundedness
+                                // has a chance to propogate to the descendant in a recursive call.
+                                nodeHolder.node.SetKnownWellFounded()
+                        } else {
+                                // This square node is not yet known to be well-founded and the child node
+                                // is not yet known to be well-founded. Set up a back link from the child.
+                                childHolder.parentsToBeNotified.Add(nodeHolder)
+                                sawUnverifiedChild = true
+                        }
+                        continue // Done handling the square case.
+                }
+
+                // Else the node is a circle. If the child is not yet known to be well-founded
+                // set up a back link from the child to this node.
+                if !childHolder.node.KnownWellFounded() {
+                        childHolder.parentsToBeNotified.Add(nodeHolder)
+                        sawUnverifiedChild = true
+                }
+        }
+
+        // If a circle node has only well-founded children, or a square node has no children at all,
+        // then the node is well-founded.
+        if !sawUnverifiedChild && !nodeHolder.node.KnownWellFounded() {
+                nodeHolder.node.SetKnownWellFounded()
+        }
+
+        // Possibly add this node to the |foundationSet| or the |pendingSet|.
+        if nodeHolder.node.KnownWellFounded() {
+                if !nodeHolder.parentsToBeNotified.Empty() {
+                        finder.foundationSet.Add(nodeHolder)
+                }
+        } else {
+                finder.pendingSet.Add(nodeHolder)
+        }
+
+        // Mark the visit as ended.
+        nodeHolder.state = vsVisitEnded
+        return
+}
+
+// checkWellFoundedPhase2 performs phase 2 of the algorithm. The goal is to
+// propogate known well-foundedness along the back-links that were established
+// during phase 1. We have two sets of nodes: the |foundationSet| and the
+// |pendingSet|. The |pendingSet| consists of all nodes that are not currently
+// known to be well-founded. If the |pendingSet| is not empty when this method
+// returns, then the nodes in the |pendingSet| are ill-founded. The algorithm
+// proceeds in rounds. The |foundationSet| consists of the current frontier of
+// the propogation. That is, the |foundationSet| consists of the nodes discovered
+// to be well-founded in the previous round. (It starts with the nodes discovered
+// to be well-founded during phase 1, pruned to nodes that have parents that
+// are in the |pendingSet|.) During each round we propogate known well-foundedness
+// from the nodes in the |foundationSet| to their parents in the |pendingSet|
+// that can now be verified as known well-foudned.
+func (finder *cycleFinder) checkWellFoundedPhase2(nodeHolder *nodeHolder) {
+        for !finder.foundationSet.Empty() {
+                nextFoundationSet := MakeNodeSet()

[azani, 2016/04/22 18:37:14]

Alternatively, you could add a Pop() method on NodeSet that returns an arbitrary
member of the set and just Pop a node each iteration until foundationSet is
empty. I think that might be slightly more readable.

[rudominer, 2016/04/23 01:22:30]

Done.

+                for n, _ := range finder.foundationSet.elements {
+                        for p, _ := range n.parentsToBeNotified.elements {
+                                if finder.pendingSet.Contains(p) {
+                                        knownWellFounded := true
+                                        if !p.node.IsSquare() {
+                                                for _, child := range p.node.OutEdges() {
+                                                        if child != p.node && !child.KnownWellFounded() {
+                                                                knownWellFounded = false
+                                                                break
+                                                        }
+                                                }
+                                        }
+                                        if knownWellFounded {
+                                                p.node.SetKnownWellFounded()
+                                                nextFoundationSet.Add(p)
+                                                finder.pendingSet.Remove(p)
+                                        }
+                                }
+                        }
+                }
+                finder.foundationSet = nextFoundationSet
+        }
+}
+
+// findKnownCycle finds and returns a |CycleDescription| starting from a node that is known
+// to be ill-founded. This proceeds by following edges from an ill-founded node to
+// an ill-founded child node until a cycle is formed. We return a *canonical* cycle,
+// meaning we start from the least possible node and follow edges to the least possible

[azani, 2016/04/22 18:37:14]

"least possible node" is ambiguous.

Also, I wonder if you might want to always return a circle first. Ultimately, we
want to tell the user: take this circle and make it a square. So maybe the
circle should be highlighted/made easy to access.

[rudominer, 2016/04/23 01:22:30]

Done
Why do you think we ultimately want to tell the user to turn his struct into a
union? There are multiple ways to avoid the ill-foundedness. Also (in the
abstract setting at least) it is possible for a graph containing only square
nodes to be ill-founded. 

+// child node. This is done in order to make testing of the algorithm easier.
+// We are not concerned with optimizing the performance of phase 3 because in the intended application
+// phase 3 can occur at most once in the lifetime of a program: Once an ill-founded node is detected the
+// program exits with a cycle descritpion allowing the user to fix the ill-foundedness.

[azani, 2016/04/22 18:37:14]

s/descritpion/description

[rudominer, 2016/04/23 01:22:30]

Done.

+func (finder *cycleFinder) findKnownCycle(nodeHolder *nodeHolder) *CycleDescription {

[azani, 2016/04/22 18:37:14]

Maybe rename findKnownIllFoundedCycle.

[rudominer, 2016/04/23 01:22:30]

Done.

+        // Mark the current node as started
+        nodeHolder.state = vsVisitStarted
+        finder.currentPath.Push(nodeHolder)
+        for _, child := range nodeHolder.node.OutEdges() {
+                childHolder := finder.holderForNode(child)
+                if childHolder.state == vsVisitStarted {
+                        // If the child has been started but not finished then we have found a cycle
+                        // from the child to the current node back to the child.
+                        return newCycleDescription(finder.currentPath.elements, childHolder, nodeHolder)
+                } else if !childHolder.node.KnownWellFounded() {
+                        return finder.findKnownCycle(childHolder)
+                }
+        }
+        panic("Should not get here.")

[azani, 2016/04/22 18:37:14]

Maybe:

"Could not find a known ill-founded cycle. That should not happen."

[rudominer, 2016/04/23 01:22:30]

Done.

+}
+
+// Returns the nodeHolder for the given node.
+func (finder *cycleFinder) holderForNode(node Node) *nodeHolder {
+        if holder, found := finder.nodeToHolder[node]; found {
+                return holder
+        }
+
+        // This is the first time we have seen this node. Assign it a new
+        // visitor order.
+        holder := newNodeHolder(node, finder.visitationIndex)
+        finder.visitationIndex++
+        finder.nodeToHolder[node] = holder
+        return holder
+}
+
+////////////////////////////////////////////////////
+// nodeHolder type
+////////////////////////////////////////////////////
+
+type visitationState int
+
+const (
+        vsUnvisited visitationState = iota
+        vsVisitStarted
+        vsVisitEnded
+)
+
+// A nodeHolder is an internal data structure used by the algorithm.
+// It holds one node plus data about that node used by the algorithm.
+type nodeHolder struct {
+        // The node
+        node Node
+
+        parentsToBeNotified NodeSet
+
+        visitationOrder int
+
+        state visitationState
+}
+
+func newNodeHolder(node Node, visitationOrder int) *nodeHolder {
+        nodeHolder := new(nodeHolder)
+        nodeHolder.node = node
+        nodeHolder.parentsToBeNotified = MakeNodeSet()
+        nodeHolder.state = vsUnvisited
+        nodeHolder.visitationOrder = visitationOrder
+        return nodeHolder
+}
+
+//////////////////////////////////////////////
+///  nodeStack type
+//////////////////////////////////////////////
+
+// A nodeStack is a stack of *nodeHolders
+type nodeStack struct {
+        elements []*nodeHolder
+}
+
+func (stack *nodeStack) Push(n *nodeHolder) {
+        stack.elements = append(stack.elements, n)
+}
+
+func (stack *nodeStack) Size() int {
+        return len(stack.elements)
+}
+
+func (stack *nodeStack) Pop() (n *nodeHolder) {
+        lastIndex := stack.Size() - 1
+        n = stack.elements[lastIndex]
+        stack.elements = stack.elements[:lastIndex]
+        return
+}
+
+func (stack *nodeStack) Peek() (n *nodeHolder) {
+        return stack.elements[stack.Size()-1]
+}
+
+func (stack *nodeStack) String() string {
+        var buffer bytes.Buffer
+        fmt.Fprintf(&buffer, "[")
+        first := true
+        for _, e := range stack.elements {
+                if !first {
+                        fmt.Fprintf(&buffer, ", ")
+                }
+                fmt.Fprintf(&buffer, "%s", e.node.DebugName())
+                first = false
+        }
+        fmt.Fprintln(&buffer, "]")
+        return buffer.String()
+}
+
+///////////////////////////////////////////////////
+/// NodeSet type
+///////////////////////////////////////////////////
+
+// A NodeSet is a set of nodeHolders.
+type NodeSet struct {
+        elements map[*nodeHolder]bool
+}
+
+// MakeNodeSet makes a new empty NodeSet.
+func MakeNodeSet() NodeSet {
+        nodeSet := NodeSet{}
+        nodeSet.elements = make(map[*nodeHolder]bool)
+        return nodeSet
+}
+
+// Add adds a Node to a NodeSet.
+func (set *NodeSet) Add(node *nodeHolder) {
+        set.elements[node] = true
+}
+
+// AddAll adds all the nodes from |otherSet| to |set|.
+func (set *NodeSet) AddAll(otherSet NodeSet) {
+        for e, _ := range otherSet.elements {
+                set.elements[e] = true
+        }
+}
+
+// Contains returns whether or not |node| is an element of |set|.
+func (set *NodeSet) Contains(node *nodeHolder) bool {
+        _, ok := set.elements[node]
+        return ok
+}
+
+func (set *NodeSet) Remove(node *nodeHolder) {
+        delete(set.elements, node)
+}
+
+func (set *NodeSet) Empty() bool {
+        return len(set.elements) == 0
+}
+
+func (set *NodeSet) ToNodeSlice() []Node {
+        slice := make([]Node, 0, len(set.elements))
+        for n, _ := range set.elements {
+                slice = append(slice, n.node)
+        }
+        return slice
+}
+
+func (set *NodeSet) Size() int {
+        return len(set.elements)
+}
+
+// compareNodeSets is a package-private method used in our tests. It returns
+// a non-nil error in case expected is not equal to actual.
+func compareNodeSets(expected, actual *NodeSet) error {
+        for n, _ := range expected.elements {
+                if !actual.Contains(n) {
+                        return fmt.Errorf("%s is in expected but not actual", n.node.DebugName())
+                }
+        }
+        for n, _ := range actual.elements {
+                if !expected.Contains(n) {
+                        return fmt.Errorf("%s is in actual but not expected", n.node.DebugName())
+                }
+        }
+        return nil
+}
+
+// String returns a human readable string representation of |set|.
+func (set *NodeSet) String() string {
+        var buffer bytes.Buffer
+        fmt.Fprintf(&buffer, "{")
+        first := true
+        for e, _ := range set.elements {
+                if !first {
+                        fmt.Fprintf(&buffer, ", ")
+                }
+                fmt.Fprintf(&buffer, "%s", e.node.DebugName())
+                first = false
+        }
+        fmt.Fprintln(&buffer, "}")
+        return buffer.String()
+}
+
+///////////////////////////////////////////////////
+// CycleDescription type
+//
+// A |CycleDescription| describes a cycle in a directed graph.
+///////////////////////////////////////////////////
+
+type CycleDescription struct {
+        first, last Node
+        path        []Node
+}
+
+func (c *CycleDescription) String() string {
+        var buffer bytes.Buffer
+        fmt.Fprintf(&buffer, "first:%s", c.first.DebugName())
+        fmt.Fprintf(&buffer, ", last:%s", c.last.DebugName())
+        fmt.Fprintf(&buffer, ", path:{")
+        first := true
+        for _, n := range c.path {
+                if !first {
+                        fmt.Fprintf(&buffer, ", ")
+                }
+                fmt.Fprintf(&buffer, "%s", n.DebugName())
+                first = false
+        }
+        fmt.Fprintln(&buffer, "}")
+        return buffer.String()
+}
+
+func newCycleDescription(path []*nodeHolder, first, last *nodeHolder) *CycleDescription {
+        description := CycleDescription{}
+        description.first = first.node
+        description.last = last.node
+        description.path = make([]Node, 0, len(path))
+        for _, n := range path {
+                if len(description.path) > 0 || n.node == first.node {
+                        description.path = append(description.path, n.node)
+                }
+        }
+        if description.path[len(description.path)-1] != last.node {
+                panic(fmt.Sprintf("%s != %s", description.path[len(description.path)-1], last.node))
+        }
+        return &description
+}