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Introduction

Several tables in the opentype format are formed internally by a graph of subtables. Parent node's reference their children through the use of positive offsets, which are typically 16 bits wide. Since offsets are always positive this forms a directed acyclic graph. For storage in the font file the graph must be given a topological ordering and then the subtables packed in serial according to that ordering. Since 16 bit offsets have a maximum value of 65,535 if the distance between a parent subtable and a child is more then 65,535 bytes then it's not possible for the offset to encode that edge.

For many fonts with complex layout rules (such as Arabic) it's not unusual for the tables containing layout rules (GSUB/GPOS) to be larger than 65kb. As a result these types of fonts are susceptible to offset overflows when serializing to the binary font format.

Offset overflows can happen for a variety of reasons and require different strategies to resolve:

Simple overflows can often be resolved with a different topological ordering.
If a subtable has many parents this can result in the link from furthest parent(s) being at risk for overflows. In these cases it's possible to duplicate the shared subtable which allows it to be placed closer to it's parent.
If subtables exist which are themselves larger than 65kb it's not possible for any offsets to point past them. In these cases the subtable can usually be split into two smaller subtables to allow for more flexibility in the ordering.
In GSUB/GPOS overflows from Lookup subtables can be resolved by changing the Lookup to an extension lookup which uses a 32 bit offset instead of 16 bit offset.

In general there isn't a simple solution to produce an optimal topological ordering for a given graph. Finding an ordering which doesn't overflow is a NP hard problem. Existing solutions use heuristics which attempt a combination of the above strategies to attempt to find a non-overflowing configuration.

The harfbuzz subsetting library includes a repacking algorithm which is used to resolve offset overflows that are present in the subsetted tables it produces. This document provides a deep dive into how the harfbuzz repacking algorithm works.

Other implementations exist, such as in fontTools, however these are not covered in this document.

Foundations

There's four key pieces to the harfbuzz approach:

Subtable Graph: a table's internal structure is abstracted out into a lightweight graph representation where each subtable is a node and each offset forms an edge. The nodes only need to know how many bytes the corresponding subtable occupies. This lightweight representation can be easily modified to test new ordering's and strategies as the repacking algorithm iterates.
Topological sorting algorithm: an algorithm which given a graph gives a linear sorting of the nodes such that all offsets will be positive.
Overflow check: given a graph and a topological sorting it checks if there will be any overflows in any of the offsets. If there are overflows it returns a list of (parent, child) tuples that will overflow. Since the graph has information on the size of each subtable it's straightforward to calculate the final position of each subtable and then check if any offsets to it will overflow.
Offset resolution strategies: given a particular occurrence of an overflow these strategies modify the graph to attempt to resolve the overflow.

High Level Algorithm

def repack(graph):
  graph.topological_sort()

  if (graph.will_overflow())
    assign_spaces(graph)
    graph.topological_sort()

  while (overflows = graph.will_overflow()):
    for overflow in overflows:
      apply_offset_resolution_strategy (overflow, graph)
    graph.topological_sort()

The actual code for this processing loop can be found in the function hb_resolve_overflows () of hb-repacker.hh.

Topological Sorting Algorithms

The harfbuzz repacker uses two different algorithms for topological sorting:

Kahn's Algorithm
Sorting by shortest distance

Kahn's algorithm is approximately twice as fast as the shortest distance sort so that is attempted first (only on the first topological sort). If it fails to eliminate overflows then shortest distance sort will be used for all subsequent topological sorting operations.

Shortest Distance Sort

This algorithm orders the nodes based on total distance to each node. Nodes with a shorter distance are ordered first.

The "weight" of an edge is the sum of the size of the sub-table being pointed to plus 2^16 for a 16 bit offset and 2^32 for a 32 bit offset.

The distance of a node is the sum of all weights along the shortest path from the root to that node plus a priority modifier (used to change where nodes are placed by moving increasing or decreasing the effective distance). Ties between nodes with the same distance are broken based on the order of the offset in the sub table bytes.

The shortest distance to each node is determined using Djikstra's algorithm. Then the topological ordering is produce by applying a modified version of Kahn's algorithm that uses a priority queue based on the shortest distance to each node.

Optimizing the Sorting

The topological sorting operation is the core of the repacker and is run on each iteration so it needs to be as fast as possible. There's a few things that are done to speed up subsequent sorting operations:

The number of incoming edges to each node is cached. This is required by the Kahn's algorithm portion of both sorts. Where possible when the graph is modified we manually update the cached edge counts of affected nodes.
The distance to each node is cached. Where possible when the graph is modified we manually update the cached distances of any affected nodes.

Caching these values allows the repacker to avoid recalculating them for the full graph on each iteration.

The other important factor to speed is a fast priority queue which is a core datastructure to the topological sorting algorithm. Currently a basic heap based queue is used. Heap based queue's don't support fast priority decreases, but that can be worked around by just adding redundant entries to the priority queue and filtering the older ones out when poppping off entries. This is based on the recommendations in a study of the practical performance of priority queues in Dijkstra's algorithm

Special Handling of 32 bit Offsets

If a graph contains multiple 32 bit offsets then the shortest distance sorting will be likely be suboptimal. For example consider the case where a graph contains two 32 bit offsets that each point to a subgraph which are not connected to each other. The shortest distance sort will interleave the subtables of the two subgraphs, potentially resulting in overflows. Since each of these subgraphs are independent of each other, and 32 bit offsets can point extremely long distances a better strategy is to pack the first subgraph in it's entirety and then have the second subgraph packed after with the 32 bit offset pointing over the first subgraph. For example given the graph:

a--- b -- d -- f
 \
  \_ c -- e -- g

Where the links from a to b and a to c are 32 bit offsets, the shortest distance sort would be:

a, b, c, d, e, f, g

If nodes d and e have a combined size greater than 65kb then the offset from d to f will overflow. A better ordering is:

a, b, d, f, c, e, g

The ability for 32 bit offsets to point long distances is utilized to jump over the subgraph of b which gives the remaining 16 bit offsets a better chance of not overflowing.

The above is an ideal situation where the subgraphs are disconnected from each other, in practice this is often not this case. So this idea can be generalized as follows:

If there is a subgraph that is only reachable from one or more 32 bit offsets, then:

That subgraph can be treated as an independent unit and all nodes of the subgraph packed in isolation from the rest of the graph.
In a table that occupies less than 4gb of space (in practice all fonts), that packed independent subgraph can be placed anywhere after the parent nodes without overflowing the 32 bit offsets from the parent nodes.

The sorting algorithm incorporates this via a "space" modifier that can be applied to nodes in the graph. By default all nodes are treated as being in space zero. If a node is given a non-zero space, n, then the computed distance to the node will be modified by adding n * 2^32. This will cause that node and it's descendants to be packed between all nodes in space n-1 and space n+1. Resulting in a topological sort like:

| space 0 subtables | space 1 subtables | .... | space n subtables |

The assign_spaces() step in the high level algorithm is responsible for identifying independent subgraphs and assigning unique spaces to each one. More information on the space assignment can be found in the next section.

Offset Resolution Strategies

Space Assignment

The goal of space assignment is to find connected subgraphs that are only reachable via 32 bit offsets and then assign each such subgraph to a unique non-zero space. The algorithm is roughly:

Collect the set, S, of nodes that are children of 32 bit offsets.
Do a directed traversal from each node in S and collect all encountered nodes into set T. Mark all nodes in the graph that are not in T as being in space 0.
Set next_space = 1.
While set S is not empty:

a. Pick a node n in set S then perform an undirected graph traversal and find the set Q of nodes that are reachable from n.

b. During traversal if a node, m, has a edge to a node in space 0 then m must be duplicated to disconnect it from space 0.

d. Remove all nodes in Q from S and assign all nodes in Q to next_space.

c. Increment next_space by one.

Manual Iterative Resolutions

For each overflow in each iteration the algorithm will attempt to apply offset overflow resolution strategies to eliminate the overflow. The type of strategy applied is dependent on the characteristics of the overflowing link:

If the overflowing offset is inside a space other than space 0 and the subgraph space has more than one 32 bit offset pointing into the subgraph then subdivide the space by moving subgraph from one of the 32 bit offsets into a new space via the duplication of shared nodes.
If the overflowing offset is pointing to a subtable with more than one incoming edge: duplicate the node so that the overflowing offset is pointing at it's own copy of that node.
Otherwise, attempt to move the child subtable closer to it's parent. This is accomplished by raising the priority of all children of the parent. Next time the topological sort is run the children will be ordered closer to the parent.

Test Cases

The harfbuzz repacker has tests defined using generic graphs: https://github.com/harfbuzz/harfbuzz/blob/main/src/test-repacker.cc

Future Improvements

The above resolution strategies are not sufficient to resolve all overflows. For example consider the case where a single subtable is 65k and the graph structure requires an offset to point over it.

The current harfbuzz implementation is suitable for the vast majority of subsetting related overflows. Subsetting related overflows are typically easy to solve since all subsets are derived from a font that was originally overflow free. A more general purpose version of the algorithm suitable for font creation purposes will likely need some additional offset resolution strategies:

Currently only children nodes are moved to resolve offsets. However, in many cases moving a parent node closer to it's children will have less impact on the size of other offsets. Thus the algorithm should use a heuristic (based on parent and child subtable sizes) to decide if the children's priority should be increased or the parent's priority decreased.
Many subtables can be split into two smaller subtables without impacting the overall functionality. This should be done when an overflow is the result of a very large table which can't be moved to avoid offsets pointing over it.
Lookup subtables in GSUB/GPOS can be upgraded to extension lookups which uses a 32 bit offset. Overflows from a Lookup subtable to it's child should be resolved by converting to an extension lookup.

Once additional resolution strategies are added to the algorithm it's likely that we'll need to switch to using a backtracking algorithm to explore the various combinations of resolution strategies until a non-overflowing combination is found. This will require the ability to restore the graph to an earlier state. It's likely that using a stack of undoable resolution commands could be used to accomplish this.

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