infra_paths: keep cache paths super short to avoid long path issues (attempt #2)

This is a reland of https://codereview.chromium.org/2102613002/ .

See https://goto.google.com/ljfzu (Google-internal) for more context. Summary:
- the only builders with chromium checkout affected by this are on chromium.fyi
- we can't clean up old cache directories before switching to new ones

Also see https://goto.google.com/oyxwm why so short directory names are OK.
Similarly https://codereview.chromium.org/2061213002 .

BUG=623575

Review-Url: https://codereview.chromium.org/2102403002

recipe_modules/infra_paths/path_config.py

@@ -36,16 +36,18 @@ def infra_kitchen(c):
     b_dir = b_dir[:-1]
 
   if c.PLATFORM in ('linux', 'mac'):
-    c.base_paths['cache'] = (
-        '/', 'b', 'cache', 'chromium')
-    for path in ('builder_cache', 'git_cache', 'goma_cache', 'goma_deps_cache'):
+    c.base_paths['cache'] = ('/', 'b', 'c')
+    c.base_paths['builder_cache'] = c.base_paths['cache'] + ('b',)
+    for path in ('git_cache', 'goma_cache', 'goma_deps_cache'):
       c.base_paths[path] = c.base_paths['cache'] + (path,)
   elif b_dir:
-    c.base_paths['cache'] = b_dir + ('cache', 'chromium')
-    for path in ('builder_cache', 'git_cache', 'goma_cache', 'goma_deps_cache'):
+    c.base_paths['cache'] = b_dir + ('c',)
+    c.base_paths['builder_cache'] = c.base_paths['cache'] + ('b',)
+    for path in ('git_cache', 'goma_cache', 'goma_deps_cache'):
       c.base_paths[path] = c.base_paths['cache'] + (path,)
   else:  # pragma: no cover
-    c.base_paths['cache'] = c.base_paths['root'] + ('cache',)
+    c.base_paths['cache'] = c.base_paths['root'] + ('c',)
+    c.base_paths['builder_cache'] = c.base_paths['cache'] + ('b',)
     c.base_paths['git_cache'] = c.base_paths['root'] + ('cache_dir',)
-    for path in ('builder_cache', 'goma_cache', 'goma_deps_cache'):
+    for path in ('goma_cache', 'goma_deps_cache'):
       c.base_paths[path] = c.base_paths['cache'] + (path,)