[SPARK-60342][CORE][WEBUI] Respect UI root in Master UI links

core/src/main/scala/org/apache/spark/deploy/master/ui/EnvironmentPage.scala

@@ -62,7 +62,7 @@ private[ui] class EnvironmentPage(
 
     val content =
       <div>
-        <p><a href="/">Back to Master</a></p>
+        <p><a href={UIUtils.prependBaseUri(request, "/")}>Back to Master</a></p>
       </div>
       <span>
         <span class="collapse-table" data-bs-toggle="collapse"

core/src/main/scala/org/apache/spark/deploy/master/ui/LogPage.scala

@@ -62,7 +62,7 @@ private[ui] class LogPage(parent: MasterWebUI) extends WebUIPage("logPage") with
     val content =
       <script type="module" src={UIUtils.prependBaseUri(request, "/static/utils.js")}></script> ++
       <div>
-        <p><a href="/">Back to Master</a></p>
+        <p><a href={UIUtils.prependBaseUri(request, "/")}>Back to Master</a></p>
         {range}
         <div class="log-content overflow-auto p-1" style="height:80vh;">
           <div>{moreButton}</div>

core/src/main/scala/org/apache/spark/deploy/master/ui/MasterPage.scala

@@ -193,8 +193,8 @@ private[ui] class MasterPage(parent: MasterWebUI) extends WebUIPage("") {
                 {state.completedDrivers.count(_.state == DriverState.RELAUNCHING)} Relaunching)
               </li>
               <li><strong>Status:</strong> {state.status}
-                (<a href={"/environment/"}>Environment</a>,
-                <a href={"/logPage/?self&logType=out"}>Log</a>)
+                (<a href={UIUtils.prependBaseUri(request, "/environment/")}>Environment</a>,
+                <a href={UIUtils.prependBaseUri(request, "/logPage/?self&logType=out")}>Log</a>)
               </li>
             </ul>
           </div>

core/src/test/scala/org/apache/spark/deploy/master/MasterSuiteBase.scala

@@ -144,6 +144,9 @@ trait MasterSuiteBase extends SparkFunSuite
   // groups represent URL and worker ID
   val WORKER_LINK_RE = """<a href="(.+?)">\s*(worker-.+?)\s*</a>""".r
 
+  // regex to extract site-relative links from the webui HTML
+  val SITE_RELATIVE_LINK_RE = """href="(/[^"]*)"""".r
+
   private var _master: Master = _
 
   after {
@@ -179,6 +182,9 @@ trait MasterSuiteBase extends SparkFunSuite
     html should include (s"""href="$proxyUrl/static""")
     html should not include ("""src="/static""")
     html should include (s"""src="$proxyUrl/static""")
+    // other site-relative links, e.g. "Back to Master", must not bypass the proxy either
+    (SITE_RELATIVE_LINK_RE findAllMatchIn html).map(_.group(1))
+      .filterNot(_.startsWith(proxyUrl)).toSeq shouldBe empty
   }
 
   protected def verifyDrivers(

core/src/test/scala/org/apache/spark/deploy/master/MasterWorkerUISuite.scala

@@ -133,6 +133,13 @@ class MasterWorkerUISuite extends MasterSuiteBase {
           .tryWithResource(Source.fromURL(s"$masterUrl/"))(_.getLines().mkString("\n"))
         html should include ("Spark Master at spark://")
         verifyStaticResourcesServedByProxy(html, reverseProxyUrl)
+        Seq("environment/", "logPage/?self&logType=out").foreach { path =>
+          val pageHtml = Utils
+            .tryWithResource(Source.fromURL(s"$masterUrl/$path"))(_.getLines().mkString("\n"))
+          withClue(s"$path: ") {
+            verifyStaticResourcesServedByProxy(pageHtml, reverseProxyUrl)
+          }
+        }
         verifyWorkerUI(html, masterUrl, reverseProxyUrl)
       }
     } finally {