# Homotopy Testing

## Multiple Obstacles

Now let’s generalize the homotopy problem from the last lecture to the case of more than one obstacle. Let $$O = \{a, b, c, \dots\}$$ be an arbitrary finite set of points in the plane. The definitions of homotopy and contractible easily generalize to polygons (and other closed curves) in $$\mathbb{R}^2\setminus O$$. How quickly can we tell whether two polygons in $$\mathbb{R}^2\setminus O$$ are homotopic?

The game Fast and Loose shows some of the subtlety of this problem. Let $$a$$ and $$b$$ be arbitrarily points in each of the two loops of the curve $$C$$ that the con artist actually uses. It’s not hard to see that $$wind(c, a) = wind(C, b) = 0$$; any ray upward from either $$a$$ or $$b$$ has one positive crossing and one negative crossing. Thus, the curve is contractible in the plane with only one of these punctures; in other words, the chain is loose if we only use one finger. But the curve $$C$$ is not contractible in $$\mathbb{R}^2 \setminus \{a, b\}$$; we can hold the chain fast by placing a finger in each loop. Winding numbers are not a homotopy invariant when there is more than one obstacle.

Just as in the one-obstacle setting, we can (approximately) decompose any homotopy between two polygons in $$\mathbb{R}^2\setminus O$$ into safe vertex moves, where now a vertex move is safe if the moving vertex and its incident edges avoids all points in $$O$$. Essentially the same argument from the previous lecture implies the following:

Theorem:
Two polygons in $$\mathbb{R}^2\setminus O$$ are homotopic if and only if they are connected by a sequence of safe vertex moves.

## Crossing Sequences

However, we can still define a homotopy invariant by shooting rays out of every obstacle, and recording how the polygon intersects these rays. Specifically, we record not only the number of times the polygon crosses each ray, but the actual order and directions of these crossings.

As usual, we will assume without loss of generality that the obstacles have distinct $$x$$-coordinates. Shoot a vertical ray upward from each obstacle point; I’ll call these vertical rays fences. The crossing sequence of a polygon $$P$$ in $$\mathbb{R}^2\setminus O$$ is the sequence of intersections between the fences and the polygon, in order along the polygon, along with the sign of each crossing (positive if the polygon crosses the fence to the left, negative if the polygon crosses the fence to the right).

Figure 2 shows a polygon in the plane with two obstacle points $$a$$ and $$b$$. If we orient the polygon as indicated by the arrows, starting at the lower left corner, the crossing sequence is BAabBAabB, where each upper-case letter denotes a positive crossing through the corresponding fence, and each lower-case letter denotes a negative crossing through the corresponding fence.

Lemma:
Two polygons in $$\mathbb{R}^2\setminus O$$ with the same crossing sequence are homotopic.
Proof:
I’ll describe a homotopy that transforms any polygon $$P$$ into a canonical polygon with the same crossing sequence. (With more care, the homotopy can be described explicitly as a sequence of safe vertex moves.)

First, we define two new sentinel points $$a^\flat$$ and $$a^\sharp$$ just above and on either side of each obstacle $$a$$. These points are close enough to $$a$$ that no vertex of $$P$$, no other obstacle, and no other sentinel point lies on or between the vertical lines through $$a^\flat$$ and $$a^\sharp$$.

Next, we divert edges through the sentinel points, by adding two additional vertices near each intersection between $$P$$ and any fence, and then moving those new vertices to the sentinel points near the corresponding obstacle.

• If the original edge crosses $$a$$’s fence from right to left (a positive crossing), the diverted edge passes through sentinel points $$a^\sharp$$ and $$a^\flat$$ in that order.
• If the original edge crosses $$a$$’s fence from left to right (a negative crossing), the diverted edge passes through sentinel points $$a^\flat$$ and $$a^\sharp$$ in that order.
See Figure 3. The resulting polygon $$P'$$ has the same crossing sequence as the original polygon $$P$$.

Finally, we divert the rest of the polygon to a single point far below any obstacle. First we add a new vertex at the midpoint of each edge of $$P'$$ between two sentinel points of different obstacles. Then we choose a point $$z$$ sufficiently far below the polygon and the sentinel points that for any polygon vertex $$p$$, the segment $$zp$$ does not cross any of the fences. Finally, we move every vertex of $$P'$$ that is not a sentinel point directly to this new bottom point $$z$$. Again, this deformation does not change the polygon’s crossing sequence.

The resulting canonical polygon $$P''$$ is a concatenation of loops of the form $$z a^\flat a^\sharp z$$ (for each negative crossing) or $$z a^\sharp a^\flat z$$ (for each positive crossing).

Corollary:
Any polygon in $$\mathbb{R}^2\setminus O$$ with an empty crossing sequence is contractible.

Unfortunately, the converses of these two results are false; a homotopy that moves one polygon vertex across one fence either inserts or deletes a pair of crossings in the polygon’s crossing sequence. Thus, crossing sequences are not homotopy invariants. Fortunately, this is the only way that a homotopy can change the crossing sequence.

## Reductions

We regard signed crossing sequences as strings of abstract symbols, where each symbol $$a$$ has a formal “inverse” $$\bar{a}$$. In our earlier example, each upper case letter is the inverse of the corresponding lower-case letter, and vice versa. Let $$x\cdot y$$ denote the concatenation of strings $$x$$ and $$y$$, and let $$\varepsilon$$ denote the empty string.

An elementary reduction is a transformation of the form $$x\cdot a\bar{a}\cdot y \mapsto x\cdot y$$, where $$x$$ and $$y$$ are (possibly empty) strings and $$a$$ is a single symbol. An elementary expansion is the reverse transformation $$x\cdot y \mapsto x\cdot a\bar{a}\cdot y$$. Two strings are equivalent if once can be transformed into the other by a sequence of elementary reductions and expansions. (Check for yourself that equivalence is in fat an equivalence relation!) We call a string trivial if it is equivalent to the empty string $$\varepsilon$$. Finally, a string is reduced if no elementary reductions are possible; for example, the empty string $$\varepsilon$$ is trivially reduced, as is any string of length $$1$$.

Crossing sequences of polygons are actually cyclic strings. Formally, a cyclic string is an equivalence class of linear strings: $(w) := \{ y\cdot x \mid x\cdot y = w \}$ For example, (ABbA) = {ABbA, BbAA, bAAB, AABb} and $$(\varepsilon) = \{\varepsilon\}$$. I’ll write $$w \sim z$$ to denote that $$w$$ is a cyclic shift of $$z$$, or equivalently $$w\in(z)$$, or equivalently $$z\in (w)$$. To emphasize that elementary reductions can “wrap around” cyclic strings, we say that a cyclic string is cyclically reduced if no elementary reductions are possible. A (cyclic) string is trivial if it is equivalent to the empty (cyclic) string.

For example, the cyclic string (EcaCbaABcEeEeAdbcCBaEdDeADCe) is trivial; two different sequences of elementary reductions are shown below (using interpuncts to represent missing symbols). In the first sequence, each elementary reduction removes the leftmost matching pair; the second sequence is more haphazard. In fact, as the following lemma implies, any sequence of elementary reductions eventually reduces this string to nothing.

EcaCbaABcEeEeAdbcCBaEdDeADCe
Ec············d··········DCe
Ec························Ce
E··························e
···························· 
EcaCbaABcEeEeAdbcCBaEdDeADCe
··a··········A··············
····························
Lemma:
Every cyclic string is equivalent to exactly one cyclically reduced cyclic string.
Proof:
Let $$w$$ be a cyclic string that allows two elementary reductions $$w\mapsto x$$ and $$w\mapsto y$$, meaning two different pairs of symbols are deleted. We claim that either $$x=y$$ or there is another string $$z$$ such that $$x\mapsto z$$ and $$y\mapsto z$$ are elementary reductions.
• If the pairs of symbols deleted by $$w\mapsto x$$ and $$w\mapsto y$$ are disjoint, then we can write $$w = (a \bar{a} \cdot w_1 \cdot c\bar{c} \cdot w_2)$$ for some (possibly empty) linear strings $$w_1$$ and $$w_2$$ and (possibly equal, possibly inverse) symbols $$a$$ and $$c$$. Without loss of generality we have $$x = (w_1 \cdot c\bar{c} \cdot w_2)$$ and $$y = (w_2 \cdot a\bar{a} \cdot w_1)$$. In this case, we can take $$z = (w_1 w_2)$$.
• If the pairs of symbols deleted by $$w\mapsto x$$ and $$w\mapsto y$$ overlap, then we can write $$w = (a \bar{a} a \cdot w')$$ for some (possibly empty) linear string $$w'$$ and some symbol $$a$$. In this case we have $$x = y = (a \cdot w')$$.

It follows that applying only elementary reductions leads to a unique reduced string; however, equivalence also allows elementary expansions. Consider two equivalent but distinct cyclic strings $$x\ne y$$, and let $$x = w_1 \leftrightarrow w_2 \leftrightarrow \cdots \leftrightarrow w_n = y$$ be a sequence of strings, each connected to its success by an elementary reduction in one direction $$w_i \mapsto w_{i+1}$$ or the other $$w_{i+1} \mapsto w_i$$.

Suppose for some index $$i$$, we have reductions $$w_i \mapsto w_{i-1}$$ and $$w_i \mapsto w_{i+1}$$. If $$w_{i-1} = w_{i+1}$$, then we can remove $$w_{i-1}$$ and $$w_i$$ to obtain a shorter transformation sequence. Otherwise, there is another string $$z_i$$ such that $$w_{i-1}\mapsto z_i$$ and $$w_{i+1} \mapsto z_i$$. Thus, by induction, we can modify our transformation sequence so that every reduction appears before every expansion.

Let $$z$$ be the shortest string in this normalized sequence. Both $$x$$ and $$y$$ can be reduced to $$z$$ using only elementary reductions. Because $$x\ne y$$, either $$x \ne z$$ or $$y\ne z$$; we conclude that at most one of $$x$$ and $$y$$ is reduced.

Lemma:
Any cyclic crossing sequence of length $$x$$ can be cyclically reduced in $$O(x)$$ time.
Proof:
The following (pseudo-)python code assumes the input X is an array of non-zero integers, with inverses indicated by negation. The algorithm runs in two phases. The first phase reduces the linear sequence $$X$$ by repeatedly removes the leftmost matching pair, using the output array as a stack. The second phase performs any remaining cyclic reductions that “wrap around” the ends of the array.
def LeftGreedyReduce(X):
n = size(X)
Y = [0] * n                             // reduced sequence = stack
top = -1                                // top stack index
// ————— linear reduction —————
for i in range(n):
if top < 0 || (X[i] != -Y[top]):    // empty or no match
top = top + 1
Y[top] = X[i]                   // push
else:
top = top - 1                   // pop
// ————— cyclic reduction —————
bot = 0
while (bot < top) and (Y[bot] = -Y[top]):
bot = bot + 1
top = top - 1
// ————— done! —————
return Y[bot:top+1]
Lemma:
Two polygons are homotopic in $$\mathbb{R}^2\setminus O$$ if and only if their crossing sequences are equivalent.
Proof (sketch):
A single safe vertex move changes the signed crossing sequence by a finite number of elementary reductions and their inverses, at most one per obstacle.

Conversely, any elementary reduction of the signed crossing sequence can be modeled by a sequence of safe vertex moves, performed either directly on the original polygon or (more easily) on the canonical polygon with the same crossing sequence.

We finally have all the ingredients of our homotopy-testing algorithm.

Theorem:
For any set $$O$$ of $$k$$ points in $$\mathbb{R}^2$$, and any two $$n$$-gons $$P$$ and $$Q$$ in $$\mathbb{R}^2\setminus O$$, we can determine whether $$P$$ and $$Q$$ are homotopic in $$\mathbb{R}^2\setminus O$$ in $$O(k\log k + kn)$$ time.
Proof (sketch):
As usual we assume without loss of generality that the obstacles and polygon vertices all have distinct $$x$$-coordinates. First we sort the obstacles from left to right in $$O(k\log k)$$ time. Then we compute the crossing sequence of $$P$$ and $$Q$$, in constant time per crossing, plus constant time per vertex. Each crossing sequence has length $$O(nk)$$. Then we cyclically reduce the two crossing sequences in $$O(nk)$$ time. Finally, we check whether the two reduced crossing sequences are equal (as cyclic strings) in linear time using Knuth-Morris-Pratt (or any other fast string-matching algorithm).

Let me emphasize here that the algorithm does not construct an explicit homotopy between the two polygons.

## Varations

To be written

• Polygons with holes: Replace each hole with a sentinel point.

• Paths:

• Paths, concatenation, reversal, loops, path homotopy
• Two paths $$\pi$$ and $$\sigma$$ are homotopic if and only if the loop $$\pi\cdot\bar\sigma$$ is contractible.
• Two polygonal paths $$P$$ and $$Q$$ are homotopic if and only if they have the same (not cyclically) reduced crossing sequence.

## …and the Aptly Named Sir Not Appearing in This Film

• alternative fences: vertical lines, spanning tree + ray
• picture hanging puzzles